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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 1

Linear Algebraic Systems - all with Video Answers

Educators

Chapter Questions

05:43

Problem 1

Solve the following systems of linear equations by reducing to triangular form and then using Back Substitution.
(a)
$$
x-y=7 \text {, }
$$
$$
x+2 y=3
$$
(b)
$$
\begin{array}{r}
6 u+v=5 \\
3 u-2 v=5
\end{array}
$$
$$
p+q-r=0, \quad 2 u-v+2 w=2,
$$
(c) $2 p-q+3 r=3$,
(d) $-u-v+3 w=1$,
$$
-p-q=6 \text {; }
$$
$$
3 u-2 w=1 ;
$$
$$
5 x_1+3 x_2-x_3=9,
$$
$x+z-2 w=-3$,
$$
3 x_1+x_2=1 \text {, }
$$
(e)
$$
\begin{aligned}
3 x_1+2 x_2-x_3 & =5 \\
x_1+x_2+x_3 & =-1
\end{aligned}
$$
(f)
$$
2 x-y+2 z-w=-5,
$$
$$
-6 y-4 z+2 w=2 \text {, }
$$
(g)
$$
\begin{aligned}
x_1+3 x_2+x_3 & =1 \\
x_2+3 x_3+x_4 & =1, \\
x_3+3 x_4 & =1 .
\end{aligned}
$$
$$
x+3 y+2 z-w=1
$$

Bobby Barnes
Bobby Barnes
University of North Texas
09:42

Problem 1

Let $A=\left(\begin{array}{rrrr}-2 & 0 & 1 & 3 \\ -1 & 2 & 7 & -5 \\ 6 & -6 & -3 & 4\end{array}\right)$
(a) What is the size of $A$ ? (b) What is its $(2,3)$ entry?
(c) $(3,1)$ entry?
(d) $1^{\text {st }}$ row? (e) $2^{\text {nd }}$ column?

Anas Venkitta
Anas Venkitta
Numerade Educator
05:20

Problem 1

Solve the following linear systems by Gaussian Elimination. (a) $\left(\begin{array}{rr}1 & -1 \\ 1 & 2\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}7 \\ 3\end{array}\right)$,
CuuDaongThanCong.com
hrips i/th com/taibeudientectit
1.3 Gaussian Elimination - Regular Case
15
(b) $\left(\begin{array}{rr}6 & 1 \\ 3 & -2\end{array}\right)\left(\begin{array}{l}u \\ v\end{array}\right)=\left(\begin{array}{l}5 \\ 5\end{array}\right)$,
(c) $\left(\begin{array}{rrr}2 & 1 & 2 \\ -1 & 3 & 3 \\ 4 & -3 & 0\end{array}\right)\left(\begin{array}{c}u \\ v \\ w\end{array}\right)=\left(\begin{array}{r}3 \\ -2 \\ 7\end{array}\right)$,
(d) $\left(\begin{array}{rrr}5 & 3 & -1 \\ 3 & 2 & -1 \\ 1 & 1 & 2\end{array}\right)\left(\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right)=\left(\begin{array}{r}9 \\ 5 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & 1 & -1 \\ 2 & -1 & 3 \\ -1 & -1 & 3\end{array}\right)\left(\begin{array}{l}p \\ q \\ r\end{array}\right)=\left(\begin{array}{l}0 \\ 3 \\ 5\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}-1 & 1 & 1 & 0 \\ 2 & -1 & 0 & 1 \\ 1 & 0 & 2 & 3 \\ 0 & 1 & -1 & -2\end{array}\right)\left(\begin{array}{l}a \\ b \\ c \\ d\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}2 & -3 & 1 & 1 \\ 1 & -1 & -2 & -1 \\ 3 & -2 & 1 & 2 \\ 1 & 3 & 2 & 1\end{array}\right)\left(\begin{array}{r}x \\ y \\ z \\ w\end{array}\right)=\left(\begin{array}{r}-1 \\ 0 \\ 5 \\ 3\end{array}\right)$.

Tony Ni
Tony Ni
Numerade Educator
05:18

Problem 1

Verify by direct multiplication that the following matrices are inverses, i.e, both
conditions in (1.37) hold: (a) $A=\left(\begin{array}{rr}2 & 3 \\ -1 & -1\end{array}\right), A^{-1}=\left(\begin{array}{rr}-1 & -3 \\ 1 & 2\end{array}\right)$;
(b) $A=\left(\begin{array}{lll}2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2\end{array}\right)$
$A^{-1}=\left(\begin{array}{rrr}3 & -1 & -1 \\ -4 & 2 & 1 \\ -1 & 0 & 1\end{array}\right)$
(c) $A=\left(\begin{array}{rrr}-1 & 3 & 2 \\ 2 & 2 & -1 \\ -2 & 1 & 3\end{array}\right), A^{-1}=\left(\begin{array}{r}-1 \\ \frac{4}{7} \\ -\frac{6}{7}\end{array}\right.$
$\left.\begin{array}{rr}1 & 1 \\ -\frac{1}{7} & -\frac{3}{7} \\ \frac{5}{7} & \frac{8}{7}\end{array}\right)$

Carole Wastog
Carole Wastog
Numerade Educator
05:18

Problem 1

Verify by direct multiplication that the following matrices are inverses, i.e., both conditions in (1.37) hold: (a) $A=\left(\begin{array}{rr}2 & 3 \\ -1 & -1\end{array}\right), A^{-1}=\left(\begin{array}{rr}-1 & -3 \\ 1 & 2\end{array}\right)$;
(b) $A=\left(\begin{array}{lll}2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2\end{array}\right)$ $A^{-1}=\left(\begin{array}{rrr}3 & -1 & -1 \\ -4 & 2 & 1 \\ -1 & 0 & 1\end{array}\right)$ (c) $A=\left(\begin{array}{rrr}-1 & 3 & 2 \\ 2 & 2 & -1 \\ -2 & 1 & 3\end{array}\right), A^{-1}=\left(\begin{array}{rrr}-1 & 1 & 1 \\ \frac{4}{7} & -\frac{1}{7} & -\frac{3}{7} \\ -\frac{6}{7} & \frac{5}{7} & \frac{8}{7}\end{array}\right)$.

Carole Wastog
Carole Wastog
Numerade Educator
01:51

Problem 1

Write down the transpose of the following matrices: (a) $\left(\begin{array}{l}1 \\ 5\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 2\end{array}\right)$,
(e) $\left(\begin{array}{lll}1 & 2 & -3\end{array}\right)$,
(f) $\left(\begin{array}{ll}1 & 2 \\ 3 & 4 \\ 5 & 6\end{array}\right)$
(g) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & 5\end{array}\right)$.

Niamat Khuda
Niamat Khuda
Numerade Educator
00:53

Problem 1

Solve the following linear systems by (i) Gaussian Elimination with Back Substitution; (ii) the Gauss-Jordan algorithm to convert the augmented matrix to the fully reduced form ( I | x ) with solution $\mathbf{x}$; (iii) computing the inverse of the coefficient matrix, and then multiplying it by the right-hand side. Keep track of the number of arithmetic operations you need to perform to complete each computation, and discuss their relative efficiency.
(a)
$$
\begin{aligned}
2 x-4 y+6 z & =6, \\
3 x+2 y & =4 \\
3 x+y & =-7, \quad \text { (b) } \quad 3 x-3 y+4 z=-1, \\
-4 x+3 y-4 z & =5,
\end{aligned}
$$
(b)
(c)
$$
x-3 y=1,
$$
$$
\begin{aligned}
3 x-7 y+5 z & =-1, \\
-2 x+6 y-5 z & =0 .
\end{aligned}
$$

James Kiss
James Kiss
Numerade Educator

Problem 1

Which of the following systems has (i) a unique solution?
(ii) infinitely many solutions? (iii) no solution? In each case, find all solutions:
(a)
$$
x-2 y=1,
$$
$$
3 x+2 y=-3 \text {. }
$$
(b)
$$
\begin{aligned}
& 2 x+y+3 z=1 \\
& x+4 y-2 z=-3
\end{aligned}
$$
$$
x+y-2 z=-3,
$$
$$
x-2 y+z=6,
$$
(c)
$$
\begin{aligned}
2 x-y+3 z=7, & \text { (d) } 2 \\
x-2 y+5 z=1 . & x \\
3 x-2 y+z & =4, \\
x+3 y-4 z & =-3, \\
\text { (f) } x-3 y+5 z & =7, \\
2 x-8 y+9 z & =10 .
\end{aligned}
$$
(d)
$$
2 x+y-3 z=-3 \text {, }
$$
$$
\begin{array}{crr}
& x-2 y+5 z=1 . & x-3 y+3 z=10 . \\
x-2 y+2 z-w=3, & 3 x-2 y+z=4, & x+2 y+1
\end{array}
$$
(f)
$$
\begin{aligned}
2 x-3 y+5 z & =7 \\
x-8 y+9 z & =10
\end{aligned}
$$
(g)
$$
\begin{aligned}
x+2 y+17 z-5 w & =50, \\
9 x-16 y+10 z-8 w & =24, \\
2 x-5 y-4 z & =-13, \\
6 x-12 y+z-4 w & =-1 .
\end{aligned}
$$

Check back soon!
03:13

Problem 1

Use Gaussian Elimination to find the determinant of the following matrices:
(a) $\left(\begin{array}{rr}2 & -1 \\ -4 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rrr}0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 10\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -2 & 1 & 3 \\ 2 & 7 & -8\end{array}\right)$
(e) $\left(\begin{array}{rrrr}5 & -1 & 0 & 2 \\ 0 & 3 & -1 & 5 \\ 0 & 0 & -4 & 2 \\ 0 & 0 & 0 & 3\end{array}\right)$,
(f)
$\left(\begin{array}{rrrr}1 & -2 & 1 & 4 \\ 2 & -4 & 0 & 0 \\ 3 & -4 & 2 & 5 \\ 0 & 2 & -4 & -9\end{array}\right)$,
(g)
$\left(\begin{array}{rrrrr}1 & -2 & 1 & 4 & -5 \\ 1 & 1 & -2 & 3 & -3 \\ 2 & -1 & -1 & 2 & 2 \\ 5 & -1 & 0 & 5 & 5 \\ 2 & 2 & 0 & 4 & -1\end{array}\right)$

Vishnu P
Vishnu P
Numerade Educator
04:16

Problem 2

How should the coefficients $a, b$, and $c$ be chosen so that the system $a x+b y+c z=3$, $a x-y+c z=1, x+b y-c z=2$ has the solution $x=1, y=2$ and $z=-1$ ?

Gaurav Kalra
Gaurav Kalra
Numerade Educator
View

Problem 2

Write down examples of (a) a $3 \times 3$ matrix; (b) a $2 \times 3$ matrix; (c) a matrix with 3 rows and 4 columns; $(d)$ a row vector with 4 entries; $(e)$ a column vector with 3 entries; (f) a matrix that is both a row vector and a column vector.

Nick Johnson
Nick Johnson
Numerade Educator
01:10

Problem 2

Write out the augmented matrix for the following linear systems. Then solve the system by first applying elementary row operations of type \#1 to place the augmented matrix in upper triangular form, followed by Back Substitution.
(a)
$$
\begin{array}{r}
x_1+7 x_2=4 \\
-2 x_1-9 x_2=2 .
\end{array}
$$
(b)
$$
\begin{aligned}
3 z-5 w & =-1 \\
2 z+w & =8 .
\end{aligned}
$$
(c)
$$
x-2 y+z=0,
$$
$$
2 y-8 z=8 \text {, }
$$
$$
-4 x+5 y+9 z=-9 \text {. }
$$
$$
p+4 q-2 r=1,
$$
$$
x_1-2 x_3=-1 \text {, }
$$
$$
-x+3 y-z+w=-2,
$$
$$
x_2-x_4=2 \text {, }
$$
(d) $\quad-2 p-3 r=-7$,
$$
3 p-2 q+2 r=-1 \text {. }
$$
(e)
$$
\begin{aligned}
x-y+3 z-w & =0, \\
y-z+4 w & =7, \\
4 x-y+z & =5 .
\end{aligned}
$$

Erik Keohane
Erik Keohane
Numerade Educator
05:05

Problem 2

Let $A=\left(\begin{array}{rrr}1 & 2 & 0 \\ 0 & 1 & 3 \\ 1 & -1 & -8\end{array}\right)$. Find the right inverse of $A$ by setting up and solving the linear system $A X=\mathrm{I}$. Verify that the resulting matrix $X$ is also a left inverse.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
05:05

Problem 2

Let $A=\left(\begin{array}{rrr}1 & 2 & 0 \\ 0 & 1 & 3 \\ 1 & -1 & -8\end{array}\right)$. Find the right inverse of $A$ by setting up and solving the linear system $A X=\mathrm{I}$. Verify that the resulting matrix $X$ is also a left inverse.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
09:51

Problem 2

Let $A=\left(\begin{array}{rrr}3 & -1 & -1 \\ 1 & 2 & 1\end{array}\right), B=\left(\begin{array}{rr}-1 & 2 \\ 2 & 0 \\ -3 & 4\end{array}\right)$. Compute $A^T$ and $B^T$. Then compute $(A B)^T$ and $(B A)^T$ without first computing $A B$ or $B A$.

Sirat Shah
Sirat Shah
Numerade Educator
02:00

Problem 2

(a) Let $A$ be an $n \times n$ matrix. Which is faster to compute, $A^2$ or $A^{-1}$ ? Justify your answer. (b) What about $A^3$ versus $A^{-1}$ ? (c) How many operations are needed to compute $A^k$ ? Hint: When $k>3$, you can get away with less than $k-1$ matrix multiplications!

Angelo Rendina
Angelo Rendina
Numerade Educator
01:04

Problem 2

Determine if the following systems are compatible and, if so, find the general solution:
(a)
$$
\begin{aligned}
& 6 x_1+3 x_2=12 \text {, } \\
& 4 x_1+2 x_2=9 \text {. } \\
& 8 x_1+12 x_2=16 \text {, } \\
& 6 x_1+9 x_2=13 \text {. } \\
&
\end{aligned}
$$
(b)
$$
\begin{aligned}
8 x_1+12 x_2 & =16 \\
6 x_1+9 x_2 & =13
\end{aligned}
$$
(d)
$$
\begin{aligned}
& 2 x_1-6 x_2+4 x_3=2, \\
& -x_1+3 x_2-2 x_3=-1 .
\end{aligned}
$$
(c)
$$
\begin{aligned}
& 2 x_1+5 x_2=2, \\
& 3 x_1+6 x_2=3 .
\end{aligned}
$$
$$
x_1+2 x_2+3 x_3+4 x_4=1,
$$
$$
2 x_1+4 x_2+6 x_3+5 x_4=0
$$
$$
3 x_1+4 x_2+x_3+x_4=0,
$$
(e)
$$
\begin{aligned}
x_2+2 x_3+8 x_4 & =7, \\
-3 x_1+x_3-7 x_4 & =9 .
\end{aligned}
$$
$$
\begin{aligned}
& 4 x_1+6 x_2+4 x_3-x_4=0 . \\
& \text { ve a common intersection: }
\end{aligned}
$$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:50

Problem 2

Verify the determinant product formula (1.85) when
$$
A=\left(\begin{array}{lll}
1 & -1 & 3 \\
2 & -1 & 1 \\
4 & -2 & 0
\end{array}\right), \quad B=\left(\begin{array}{rrr}
0 & 1 & -1 \\
1 & -3 & -2 \\
2 & 0 & 1
\end{array}\right)
$$

Caleb Wood
Caleb Wood
Numerade Educator
02:24

Problem 3

The system $2 x=-6,-4 x+3 y=3, x+4 y-z=7$, is in lower triangular form.
(a) Formulate a method of Forward Substitution to solve it. (b) What happens if you reduce the system to (upper) triangular form using the algorithm in this section?
(c) Devise an algorithm that uses our linear system operation to reduce a system to lower triangular form and then solve it by Forward Substitution. (d) Check your algorithm by applying it to one or two of the systems in Exercise 1.1.1. Are you able to solve them in all cases?

Amy Jiang
Amy Jiang
Numerade Educator
04:17

Problem 3

For which values of $x, y, z, w$ are the matrices $\left(\begin{array}{cc}x+y & x-z \\ y+w & x+2 w\end{array}\right)$ and $\left(\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right)$ equal?

Sanchit Jain
Sanchit Jain
Numerade Educator
03:41

Problem 3

For each of the following augmented matrices write out the corresponding linear system of equations. Solve the system by applying Gaussian Elimination to the augmented matrix.
(a) $\left(\begin{array}{rr|r}3 & 2 & 2 \\ -4 & -3 & -1\end{array}\right)$,
(b) $\left(\begin{array}{rrr|r}1 & 2 & 0 & -3 \\ -1 & 2 & 1 & -6 \\ -2 & 0 & -3 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrrr|r}2 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 1 \\ 0 & -1 & 2 & -1 & 1 \\ 0 & 0 & -1 & 2 & 0\end{array}\right)$.

AG
Ankit Gupta
Numerade Educator
06:55

Problem 3

Write down the inverse of each of the following elementary matrices: (a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 0 \\ 5 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 6 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$,
(f) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$.

Mohan Jain
Mohan Jain
Numerade Educator
06:55

Problem 3

Write down the inverse of each of the following elementary matrices: (a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 0 \\ 5 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 6 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$,
(f) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$.

Mohan Jain
Mohan Jain
Numerade Educator
01:49

Problem 3

Show that $(A B)^T=A^T B^T$ if and only if $A$ and $B$ are square commuting matrices.

WM
William Mead
Numerade Educator
01:58

Problem 3

Which is faster: Back Substitution or multiplying a matrix by a vector? How much faster?

Neel Faucher
Neel Faucher
Numerade Educator
03:20

Problem 3

Graph the following planes and determine whether they have a common intersection:
$$
x+y+z=1, \quad x+y=1, \quad x+z=1 .
$$

Nick Johnson
Nick Johnson
Numerade Educator
02:32

Problem 3

(a) Give an example of a non-diagonal $2 \times 2$ matrix for which $A^2=\mathrm{I}$. (b) In general, if $A^2=\mathrm{I}$, show that $\operatorname{det} A= \pm 1$. (c) If $A^2=A$, what can you say about $\operatorname{det} A$ ?

Manisha Sarker
Manisha Sarker
Numerade Educator
03:38

Problem 4

For each of the systems in Exercise 1.1.1, write down the coefficient matrix $A$ and the vectors $\mathbf{x}$ and $\mathbf{b}$.

Smita Praharaj
Smita Praharaj
Numerade Educator
09:03

Problem 4

Which of the following matrices are regular?
(a) $\left(\begin{array}{ll}2 & 1 \\ 1 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & -1 \\ 3 & -2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}3 & -2 & 1 \\ -1 & 4 & -3 \\ 3 & -2 & 5\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & -2 & 3 \\ -2 & 4 & -1 \\ 3 & -1 & 2\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 3 & -3 & 0 \\ -1 & 0 & -1 & 2 \\ 3 & 3 & -6 & 1 \\ 2 & 3 & -3 & 5\end{array}\right)$

AB
Aqib Basheer
Numerade Educator
03:33

Problem 4

Show that the inverse of $L=\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & 0 & 1\end{array}\right)$ is $L^{-1}=\left(\begin{array}{rrr}1 & 0 & 0 \\ -a & 1 & 0 \\ -b & 0 & 1\end{array}\right)$. However, the inverse of $M=\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1\end{array}\right)$ is not $\left(\begin{array}{rrr}1 & 0 & 0 \\ -a & 1 & 0 \\ -b & -c & 1\end{array}\right)$. What is $M^{-1}$ ?

David Mccaslin
David Mccaslin
Numerade Educator
03:33

Problem 4

Show that the inverse of $L=\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & 0 & 1\end{array}\right)$ is $L^{-1}=\left(\begin{array}{rrr}1 & 0 & 0 \\ -a & 1 & 0 \\ -b & 0 & 1\end{array}\right)$. However, the inverse of $M=\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1\end{array}\right)$ is not $\left(\begin{array}{rrr}1 & 0 & 0 \\ -a & 1 & 0 \\ -b & -c & 1\end{array}\right)$. What is $M^{-1}$ ?

David Mccaslin
David Mccaslin
Numerade Educator

Problem 4

Prove formula (1.55).

Check back soon!
02:29

Problem 4

Use induction to prove the summation formulas (1.62), (1.63) and (1.64).

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:26

Problem 4

Let $A=\left(\begin{array}{ccc|c}a & 0 & b & 2 \\ a & 2 & a & b \\ b & 2 & a & a\end{array}\right)$ be the augmented matrix for a linear system. For which values of $a$ and $b$ does the system have (i) a unique solution? (ii) infinitely many solutions? (iii) no solution?

AG
Ankit Gupta
Numerade Educator
17:09

Problem 4

True or false: If true, explain why. If false, give an explicit counterexample.
(a) If $\operatorname{det} A \neq 0$ then $A^{-1}$ exists. (b) $\operatorname{det}(2 A)=2 \operatorname{det} A$. (c) $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$.
(d) $\operatorname{det} A^{-T}=\frac{1}{\operatorname{det} A}$.
(e) $\operatorname{det}\left(A B^{-1}\right)=\frac{\operatorname{det} A}{\operatorname{det} B}$.
(f) $\operatorname{det}[(A+B)(A-B)]=\operatorname{det}\left(A^2-B^2\right)$.
(g) If $A$ is an $n \times n$ matrix with $\operatorname{det} A=0$, then $\operatorname{rank} A<n$.
(h) If $\operatorname{det} A=1$ and $A B=\mathrm{O}$, then $B=\mathrm{O}$.

Sirat Shah
Sirat Shah
Numerade Educator
02:57

Problem 5

Write out and solve the linear systems corresponding to the indicated matrix, vector of unknowns, and right-hand side,
(a) $A=\left(\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right), \mathbf{x}=\left(\begin{array}{l}x \\ y\end{array}\right), \mathbf{b}=\left(\begin{array}{l}-1 \\ -3\end{array}\right)$;
(b) $A=\left(\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right), \mathbf{x}=\left(\begin{array}{c}u \\ v \\ w\end{array}\right), \mathbf{b}=\left(\begin{array}{r}-1 \\ -1 \\ 2\end{array}\right)$;
(c) $A=\left(\begin{array}{rrr}3 & 0 & -1 \\ -2 & -1 & 0 \\ 1 & 1 & -3\end{array}\right)$,
$\mathbf{x}=\left(\begin{array}{c}x_1 \\ x_2 \\ x_3\end{array}\right), \quad \mathbf{b}=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$
d) $A=\left(\begin{array}{rrrr}1 & 1 & -1 & -1 \\ -1 & 0 & 1 & 2 \\ 1 & -1 & 1 & 0 \\ 0 & 2 & -1 & 1\end{array}\right), \quad \mathbf{x}=\left(\begin{array}{c}x \\ y \\ z \\ w\end{array}\right), \quad \mathbf{b}=\left(\begin{array}{l}0 \\ 4 \\ 1 \\ 5\end{array}\right)$.

AG
Ankit Gupta
Numerade Educator
04:05

Problem 5

The techniques that are developed for solving linear systems are also applicable to systems with complex coefficients, whose solutions may also be complex. Use Gaussian Elimination to solve the following complex linear systems.
(a)
$$
\begin{aligned}
-\mathrm{i} x_1+(1+\mathrm{i}) x_2 & =-1 \\
(1-\mathrm{i}) x_1+x_2 & =-3 \mathrm{i} .
\end{aligned}
$$
$$
\mathrm{i} x+(1-\mathrm{i}) z=2 \mathrm{i},
$$
(b)
$$
\begin{aligned}
2 \mathrm{i} y+(1+\mathrm{i}) z=2 \\
-x+2 \mathrm{i} y+\mathrm{i} z=1-2 \mathrm{i} \\
(1+\mathrm{i}) x+\mathrm{i} y+(2+2 \mathrm{i}) z=0 \\
(1-\mathrm{i}) x+2 y+\mathrm{i} z=0 \\
(3-3 \mathrm{i}) x+\mathrm{i} y+(3-11 \mathrm{i}) z=6 .
\end{aligned}
$$
(c)
$$
\begin{aligned}
(1-\mathrm{i}) x+2 y & =\mathrm{i}, \\
-\mathrm{i} x+(1+\mathrm{i}) y & =-1 .
\end{aligned}
$$
(d)
$$
(3-3 \mathrm{i}) x+\mathrm{i} y+(3-11 \mathrm{i}) z=6 .
$$

Vishal Parmar
Vishal Parmar
Numerade Educator
05:08

Problem 5

Explain why a matrix with a row of all zeros does not have an inverse.

AG
Ankit Gupta
Numerade Educator
05:08

Problem 5

Explain why a matrix with a row of all zeros does not have an inverse.

AG
Ankit Gupta
Numerade Educator
01:19

Problem 5

Find a formula for the transposed product $(A B C)^T$ in terms of $A^T, B^T$ and $C^T$.

Derrick Hanson
Derrick Hanson
Numerade Educator
06:58

Problem 5

Let $A$ be a general $n \times n$ matrix. Determine the exact number of arithmetic operations needed to compute $A^{-1}$ using (a) Gaussian Elimination to factor $P A=L U$ and then Forward and Back Substitution to solve the $n$ linear systems (1.65); (b) the GaussJordan method. Make sure your totals do not count adding or subtracting a known 0, or multiplying or dividing by a known \pm 1 .

Matthias Takele
Matthias Takele
Numerade Educator
07:53

Problem 5

Determine the general (complex) solution to the following systems:
(a)
$$
\begin{gathered}
2 x+(1+\mathrm{i}) y-2 \mathrm{i} z=2 \mathrm{i} \\
(1-\mathrm{i}) x+y-2 \mathrm{i} z=0 .
\end{gathered}
$$
$$
x+2 \mathrm{i} y+(2-4 \mathrm{i}) z=5+5 \mathrm{i},
$$
(b)
$$
\begin{aligned}
& (-1+\mathrm{i}) x+2 y+(4+2 \mathrm{i}) z=0 \\
& (1-\mathrm{i}) x+(1+4 \mathrm{i}) y-5 \mathrm{i} z=10+5 \mathrm{i} .
\end{aligned}
$$

Christine Anacker
Christine Anacker
Numerade Educator
02:24

Problem 5

Prove that the similar matrices $B=S^{-1} A S$ have the same determinant: $\operatorname{det} A=\operatorname{det} B$.

Nick Johnson
Nick Johnson
Numerade Educator
00:58

Problem 6

(a) Write down the $5 \times 5$ identity and zero matrices.
(b) Write down their sum and their product. Does the order of multiplication matter?

AG
Ankit Gupta
Numerade Educator
01:10

Problem 6

(a) Write down an example of a system of 5 linear equations in 5 unknowns with regular diagonal coefficient matrix. (b) Solve your system. (c) Explain why solving a system whose coefficient matrix is diagonal is very easy.
1.3.7. Find the equation of the parabola $y=a x^2+b x+c$ that goes through the points $(1,6),(2,4)$, and $(3,0)$.
1.3.8. A linear system is called homogeneous if all the right-hand sides are zero, and so takes the matrix form $A \mathbf{x}=\mathbf{0}$. Explain why the solution to a homogeneous system with regular coefficient matrix is $\mathbf{x}=\mathbf{0}$.

Thomas Emment
Thomas Emment
Numerade Educator
09:22

Problem 6

(a) Write down the inverse of the matrices $A=\left(\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}1 & -1 \\ 1 & 2\end{array}\right)$. (b) Write down the product matrix $C=A B$ and its inverse $C^{-1}$ using the inverse product formula.

Kira Schwander
Kira Schwander
Numerade Educator
09:22

Problem 6

(a) Write down the inverse of the matrices $A=\left(\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}1 & -1 \\ 1 & 2\end{array}\right)$. (b) Write down the product matrix $C=A B$ and its inverse $C^{-1}$ using the inverse product formula.

Kira Schwander
Kira Schwander
Numerade Educator
01:12

Problem 6

True or false: Every square matrix $A$ commutes with its transpose $A^T$.

Vysakh M
Vysakh M
Numerade Educator
03:34

Problem 6

Count the number of arithmetic operations needed to solve a system the "old-fashioned" way, by using elementary row operations of all three types, in the same order as the GaussJordan scheme, to fully reduce the augmented matrix $M=(A \mid \mathbf{b})$ to the form ( I $\mid \mathbf{d})$, with $\mathbf{x}=\mathbf{d}$ being the solution.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
01:25

Problem 6

For which values of $b$ and $c$ does the system $x_1+x_2+b x_3=1, b x_1+3 x_2-x_3=-2$, $3 x_1+4 x_2+x_3=c$, have (a) no solution? (b) exactly one solution?
(c) infinitely many solutions?

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 6

Prove that if $A$ is a $n \times n$ matrix and $c$ is a scalar, then $\operatorname{det}(c A)=c^n \operatorname{det} A$.

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13:14

Problem 7

Consider the matrices $A=\left(\begin{array}{rrr}1 & -1 & 3 \\ -1 & 4 & -2 \\ 3 & 0 & 6\end{array}\right), B=\left(\begin{array}{rrr}-6 & 0 & 3 \\ 4 & 2 & -1\end{array}\right), C=\left(\begin{array}{rr}2 & 3 \\ -3 & -4 \\ 1 & 2\end{array}\right)$. Compute the indicated combinations where possible. (a) $3 A-B$, (b) $A B$, (c) $B A$, (d) $(A+B) C$, (e) $A+B C$, (f) $A+2 C B$, (g) $B C B-\mathrm{I}$, (h) $A^2-3 A+\mathrm{I}$, (i) $(B-\mathrm{I})(C+\mathrm{I})$.

Carole Wastog
Carole Wastog
Numerade Educator
04:44

Problem 7

Find the equation of the parabola $y=a x^2+b x+c$ that goes through the points $(1,6),(2,4)$, and $(3,0)$.

James Kiss
James Kiss
Numerade Educator
02:15

Problem 7

(a) Find the inverse of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$, where $\theta \in \mathbb{R}$.
(b) Use your result to solve the system $x=a \cos \theta-b \sin \theta, y=a \sin \theta+b \cos \theta$, for $a$ and $b$ in terms of $x$ and $y$. (c) Prove that, for all $a \in \mathbb{R}$ and $0<\theta<\pi$, the matrix $R_\theta-a$ I has an inverse.

Gregory Higby
Gregory Higby
Numerade Educator
02:15

Problem 7

(a) Find the inverse of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$, where $\theta \in \mathbb{R}$.
(b) Use your result to solve the system $x=a \cos \theta-b \sin \theta, y=a \sin \theta+b \cos \theta$, for $a$ and $b$ in terms of $x$ and $y$. (c) Prove that, for all $a \in \mathbb{R}$ and $0<\theta<\pi$, the matrix $R_\theta-a$ I has an inverse.

Gregory Higby
Gregory Higby
Numerade Educator
03:34

Problem 7

A square matrix is called normal if it commutes with its transpose: $A^T A=A A^T$. Find all normal $2 \times 2$ matrices.

Nick Johnson
Nick Johnson
Numerade Educator
04:46

Problem 7

An alternative solution strategy, also called Gauss-Jordan in some texts, is, once a pivot is in position, to use elementary row operations of type \#1 to eliminate all entries both above and below it, thereby reducing the augmented matrix to diagonal form $(D \mid \mathbf{c})$ where $D=\operatorname{diag}\left(d_1, \ldots, d_n\right)$ is a diagonal matrix containing the pivots. The solutions $x_i=c_i / d_i$ are then obtained by simple division. Is this strategy more efficient, less efficient, or the same as Gaussian Elimination with Back Substitution? Justify your answer with an exact operations count.

Jingyun Wang
Jingyun Wang
Numerade Educator
04:19

Problem 7

Determine the rank of the following matrices:
(c) $\left(\begin{array}{rrr}1 & -1 & 1 \\ 1 & -1 & 2 \\ -1 & 1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & -1 & 0 \\ 2 & -1 & 1 \\ 1 & 1 & -1\end{array}\right)$,
(g) $\left(\begin{array}{rr}0 & -3 \\ 4 & -1 \\ 1 & 2 \\ -1 & -5\end{array}\right)$,
(a) $\left(\begin{array}{rr}1 & 1 \\ 1 & -2\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & 1 & 3 \\ -2 & -1 & -3\end{array}\right)$,
(e) $\left(\begin{array}{r}3 \\ 0 \\ -2\end{array}\right)$,
$(f)$
$\left(\begin{array}{llll}0 & -1 & 2 & 5\end{array}\right)$,
$\left(\begin{array}{rrrr}1 & -1 & 2 & 1 \\ 2 & 1 & -1 & 0 \\ 1 & 2 & -3 & -1 \\ 4 & -1 & 3 & 2 \\ 0 & 3 & -5 & -2\end{array}\right)$,
(i) $\left(\begin{array}{rrrrr}0 & 0 & 0 & 3 & 1 \\ 1 & 2 & -3 & 1 & -2 \\ 2 & 4 & -2 & 1 & -2\end{array}\right)$.
(i) $\left(\begin{array}{rrrrr}0 & 0 & 0 & 3 & 1 \\ 1 & 2 & -3 & 1 & -2 \\ 2 & 4 & -2 & 1 & -2\end{array}\right)$.

Supratim Pal
Supratim Pal
Numerade Educator

Problem 7

Prove that the determinant of a lower triangular matrix is the product of its diagonal entries.

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Problem 8

Which of the following pairs of matrices commute under matrix multiplication?
(a) $\left(\begin{array}{rr}1 & 2 \\ -2 & 1\end{array}\right),\left(\begin{array}{ll}2 & 3 \\ 5 & 0\end{array}\right)$,
(b)
$$
\left(\begin{array}{rr}
3 & -1 \\
0 & 2 \\
1 & 4
\end{array}\right),\left(\begin{array}{rrr}
4 & 2 & -2 \\
5 & 2 & 4
\end{array}\right)
$$
(c)
$\left(\begin{array}{rrr}3 & 0 & -1 \\ -2 & -1 & 2 \\ 2 & 0 & 0\end{array}\right),\left(\begin{array}{lll}2 & 0 & -1 \\ 1 & 1 & -1 \\ 2 & 0 & -1\end{array}\right)$

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02:16

Problem 8

A linear system is called homogeneous if all the right-hand sides are zero, and so takes the matrix form $A \mathbf{x}=\mathbf{0}$. Explain why the solution to a homogeneous system with regular coefficient matrix is $\mathbf{x}=\mathbf{0}$.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:32

Problem 8

(a) Write down the inverses of each of the $3 \times 3$ permutation matrices (1.30). (b) Which ones are their own inverses, $P^{-1}=P$ ? $(c)$ Can you find a non-elementary permutation matrix $P$ that is its own inverse: $P^{-1}=P$ ?

Victor Salazar
Victor Salazar
Numerade Educator
02:32

Problem 8

(a) Write down the inverses of each of the $3 \times 3$ permutation matrices (1.30). (b) Which ones are their own inverses, $P^{-1}=P$ ? (c) Can you find a non-elementary permutation matrix $P$ that is its own inverse: $P^{-1}=P$ ?

Victor Salazar
Victor Salazar
Numerade Educator
04:03

Problem 8

(a) Prove that the inverse transpose operation (1.57) respects matrix multiplication: $(A B)^{-T}=A^{-T} B^{-T}$.
(b) Verify this identity for $A=\left(\begin{array}{rr}1 & -1 \\ 1 & 0\end{array}\right), B=\left(\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right)$.

Mauricio Araiza Canizales
Mauricio Araiza Canizales
Numerade Educator
03:52

Problem 8

Here, we describe a remarkable algorithm for matrix multiplication discovered by Strassen, [82]. Let $A=\left(\begin{array}{ll}A_1 & A_2 \\ A_3 & A_4\end{array}\right), B=\left(\begin{array}{ll}B_1 & B_2 \\ B_3 & B_4\end{array}\right)$, and $C=\left(\begin{array}{ll}C_1 & C_2 \\ C_3 & C_4\end{array}\right)=A B$ be block matrices of size $n=2 m$, where all blocks are of size $m \times m$. (a) Let $D_1=$ $\left(A_1+A_4\right)\left(B_1+B_4\right), D_2=\left(A_1-A_3\right)\left(B_1+B_2\right), D_3=\left(A_2-A_4\right)\left(B_3+B_4\right)$, $D_4=\left(A_1+A_2\right) B_4, D_5=\left(A_3+A_4\right) B_1, D_6=A_4\left(B_1-B_3\right), D_7=A_1\left(B_2-B_4\right)$. Show that $C_1=D_1+D_3-D_4-D_6, C_2=D_4+D_7, C_3=D_5-D_6, C_4=D_1-D_2-D_5+D_7$.
(b) How many arithmetic operations are required when $A$ and $B$ are $2 \times 2$ matrices? How does this compare with the usual method of multiplying $2 \times 2$ matrices?
(c) In the general case, suppose we use standard matrix multiplication for the matrix products in $D_1, \ldots, D_7$. Prove that Strassen's Method is faster than the direct algorithm for computing $A B$ by a factor of $\approx \frac{7}{8}$. (d) When $A$ and $B$ have size $n \times n$ with $n=2^r$, we can recursively apply Strassen's Method to multiply the $2^{r-1} \times 2^{r-1}$ blocks $A_i, B_i$. Prove that the resulting algorithm requires a total of $7^r=n^{\log _2 7}=n^{2.80735}$ multiplications

Faizanullah Kazmi
Faizanullah Kazmi
Numerade Educator

Problem 8

Write out a $P A=L U$ factorization for each of the matrices in Exercise 1.8.7.

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02:00

Problem 8

(a) Show that if $A$ has size $n \times n$, then $\operatorname{det}(-A)=(-1)^n \operatorname{det} A$. (b) Prove that, for $n$ odd, any $n \times n$ skew-symmetric matrix $A=-A^T$ is singular. (c) Find a nonsingular skew-symmetric matrix.

Robert Daugherty
Robert Daugherty
Numerade Educator
02:21

Problem 9

List the diagonal entries of $A=\left(\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16\end{array}\right)$.

Carole Wastog
Carole Wastog
Numerade Educator
00:34

Problem 9

Under what conditions do two $2 \times 2$ upper triangular matrices commute?

Thomas Emment
Thomas Emment
Numerade Educator
05:27

Problem 9

Find the inverse of the following permutation matrices:
(a) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$,
(b)
$\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$
(c)
$\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$
(d)
$\left(\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\end{array}\right)$.

William Scherer
William Scherer
Numerade Educator
05:27

Problem 9

Find the inverse of the following permutation matrices:
(a) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$,
(b) $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$
(c) $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$
(d) $\left(\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\end{array}\right)$

William Scherer
William Scherer
Numerade Educator
04:09

Problem 9

Prove that if $A$ is an invertible matrix, then $A A^T$ and $A^T A$ are also invertible.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:07

Problem 9

For each of the following tridiagonal systems find the $L U$ factorization of the coefficient matrix, and then solve the system.
(a) $\left(\begin{array}{rrr}1 & 2 & 0 \\ -1 & -1 & 1 \\ 0 & -2 & 3\end{array}\right) \mathbf{x}=\left(\begin{array}{r}4 \\ -1 \\ -6\end{array}\right)$,
(b) $\left(\begin{array}{rrrr}1 & -1 & 0 & 0 \\ -1 & 2 & 1 & 0 \\ 0 & -1 & 4 & 1 \\ 0 & 0 & -5 & 6\end{array}\right) \mathbf{X}=\left(\begin{array}{l}1 \\ 0 \\ 6 \\ 7\end{array}\right)$,
(c)
$$
\left(\begin{array}{rrrr}
1 & 2 & 0 & 0 \\
-1 & -3 & 0 & 0 \\
0 & -1 & 4 & -1 \\
0 & 0 & -1 & -1
\end{array}\right) \mathbf{x}=\left(\begin{array}{r}
0 \\
-2 \\
-3 \\
1
\end{array}\right)
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:27

Problem 9

Construct a system of three linear equations in three unknowns that has (a) one and only one solution; (b) more than one solution; (c) no solution.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:10

Problem 9

Prove directly that the $2 \times 2$ determinant formula (1.38) satisfies the four determinant axioms listed in Theorem 1.50.

Carson Merrill
Carson Merrill
Numerade Educator
01:34

Problem 10

Write out the following diagonal matrices: (a) diag $(1,0,-1)$, (b) diag $(2,-2,3,-3)$.

Carole Wastog
Carole Wastog
Numerade Educator
00:19

Problem 10

A matrix is called lower triangular if all entries above the diagonal are zero. Show that a matrix is both lower and upper triangular if and only if it is a diagonal matrix.

Fuzail Shakir
Fuzail Shakir
Numerade Educator

Problem 10

Explain how to write down the inverse permutation using the notation of Exercise 1.4.17. Apply your method to the examples in Exercise 1.5.9, and check the result by verifying that it produces the inverse permutation matrix.

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Problem 10

Explain how to write down the inverse permutation using the notation of Exercise 1.4.17. Apply your method to the examples in Exercise 1.5.9, and check the result by verifying that it produces the inverse permutation matrix.

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01:24

Problem 10

If $\mathbf{v}, \mathbf{w}$ are column vectors with the same number of entries, does $\mathbf{v} \mathbf{w}^T=\mathbf{w} \mathbf{v}^T$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
03:05

Problem 10

True or false: (a) The product of two tridiagonal matrices is tridiagonal.
(b) The inverse of a tridiagonal matrix is tridiagonal.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
02:26

Problem 10

Find a coefficient matrix $A$ such that the associated linear system $A \mathbf{x}=\mathbf{b}$ has
(a) infinitely many solutions for every $\mathbf{b}$; (b) 0 or $\infty$ solutions, depending on $\mathbf{b}$;
(c) 0 or 1 solution depending on $\mathbf{b}$; (d) exactly 1 solution for all $\mathbf{b}$.

AG
Ankit Gupta
Numerade Educator
01:47

Problem 10

In this exercise, we prove the determinantal product formula (1.85). (a) Prove that if $E$ is any elementary matrix (of the appropriate size), then $\operatorname{det}(E B)=\operatorname{det} E \operatorname{det} B$. (b) Use induction to prove that if $A=E_1 E_2 \cdots E_N$ is a product of elementary matrices, then $\operatorname{det}(A B)=\operatorname{det} A \operatorname{det} B$. Explain why this proves the product formula whenever $A$ is a nonsingular matrix. (c) Prove that if $Z$ is a matrix with a zero row, then $Z B$ also has a zero row, and so $\operatorname{det}(Z B)=0=\operatorname{det} Z \operatorname{det} B$. (d) Use Exercise 1.8.21 to complete the proof of the product formula.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
03:33

Problem 11

True or false: (a) The sum of two diagonal matrices of the same size is a diagonal matrix. (b) The product is also diagonal.

Patrick Burns
Patrick Burns
Numerade Educator
00:19

Problem 11

A square matrix is called strictly lower (upper) triangular if all entries on or above (below) the main diagonal are 0 . (a) Prove that every square matrix can be uniquely written as a sum $A=L+D+U$, with $L$ strictly lower triangular, $D$ diagonal, and $U$ strictly upper triangular.
(b) Decompose $A=\left(\begin{array}{rrr}3 & 1 & -1 \\ 1 & -4 & 2 \\ -2 & 0 & 5\end{array}\right)$ in this manner.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
09:44

Problem 11

Find all real $2 \times 2$ matrices that are their own inverses: $A^{-1}=A$.

Sirat Shah
Sirat Shah
Numerade Educator
09:44

Problem 11

Find all real $2 \times 2$ matrices that are their own inverses: $A^{-1}=A$.

Sirat Shah
Sirat Shah
Numerade Educator
02:33

Problem 11

Is there a matrix analogue of formula (1.56), namely $A^T B=B^T A$ ?

Chris Trentman
Chris Trentman
Numerade Educator
View

Problem 11

(a) Find the $L U$ factorization of the $n \times n$ tridiagonal matrix $A_n$ with all 2's along the diagonal and all -1 's along the sub- and super-diagonals for $n=3,4$, and 5 .
(b) Use your factorizations to solve the system $A_n \mathbf{x}=\mathbf{b}$, where $\mathbf{b}=(1,1,1, \ldots, 1)^T$.
(c) Can you write down the $L U$ factorization of $A_n$ for general $n$ ? Do the entries in the factors approach a limit as $n$ gets larger and larger? (d) Can you find the solution to the system $A_n \mathbf{x}=\mathbf{b}=(1,1,1, \ldots, 1)^T$ for general $n$ ?

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 11

Give an example of a nonlinear system of two equations in two unknowns that has
(a) no solution; (b) exactly two solutions; (c) exactly three solutions; (d) infinitely many solutions.

Charles Carter
Charles Carter
Numerade Educator

Problem 11

Prove (1.86).

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View

Problem 12

(a) Show that if $D=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right)$ is a $2 \times 2$ diagonal matrix with $a \neq b$, then the only matrices that commute (under matrix multiplication) with $D$ are other $2 \times 2$ diagonal matrices. (b) What if $a=b$ ? (c) Find all matrices that commute with $D=\left(\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right)$, where $a, b, c$ are all different. (d) Answer the same question for the case when $a \neq b=c$. (e) Prove that a matrix $A$ commutes with an $n \times n$ diagonal matrix $D$ with all distinct diagonal entries if and only if $A$ is a diagonal matrix.

Victor Salazar
Victor Salazar
Numerade Educator
03:19

Problem 12

A square matrix $N$ is called nilpotent if $N^k=\mathrm{O}$ for some $k \geq 1$.
(a) Show that $N=\left(\begin{array}{lll}0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$ is nilpotent. (b) Show that every strictly upper triangular matrix, as defined in Exercise 1.3.11, is nilpotent. (c) Find a nilpotent matrix which is neither lower nor upper triangular.

Joe Lesueur
Joe Lesueur
Numerade Educator
02:12

Problem 12

Construct a multiplication table that shows all possible products of the $3 \times 3$ permutation matrices (1.30). List all pairs that commute.

James Kiss
James Kiss
Numerade Educator
02:26

Problem 12

Show that if a square matrix $A$ satisfies $A^2-3 A+\mathrm{I}=\mathrm{O}$, then $A^{-1}=3 \mathrm{I}-A$.

Thomas Emment
Thomas Emment
Numerade Educator
02:13

Problem 12

(a) Let $A$ be an $m \times n$ matrix. Let $\mathbf{e}_j$ denote the $1 \times n$ column vector with a single 1 in the $j^{\text {th }}$ entry, as in (1.44). Explain why the product $A \mathbf{e}_j$ equals the $j^{\text {th }}$ column of $A$. (b) Similarly, let $\hat{\mathbf{e}}_i$ be the $1 \times m$ column vector with a single 1 in the $i^{\text {th }}$ entry. Explain why the triple product $\hat{\mathbf{e}}_i^T A \mathbf{e}_j=a_{i j}$ equals the $(i, j)$ entry of the matrix $A$.

Patrick Burns
Patrick Burns
Numerade Educator
01:08

Problem 12

Answer Exercise 1.7.11 if the super-diagonal entries of $A_n$ are changed to +1 .

Jacob Denson
Jacob Denson
Numerade Educator
02:17

Problem 12

What does it mean if a linear system has a coefficient matrix with a column of all 0's?

Nicholas Barvinok
Nicholas Barvinok
Numerade Educator
06:13

Problem 12

Prove (1.89).

Mohan Jain
Mohan Jain
Numerade Educator
03:06

Problem 13

Show that the matrix products $A B$ and $B A$ have the same size if and only if $A$ and $B$ are square matrices of the same size.

Chandra Jain
Chandra Jain
Numerade Educator
06:22

Problem 13

A square matrix $W$ is called unipotent if $N=W-\mathrm{I}$ is nilpotent, as in Exercise 1.3.12, so $(W-\mathrm{I})^k=\mathrm{O}$ for some $k \geq 1$. (a) Show that every lower or upper triangular matrix is unipotent if and only if it is unitriangular, meaning its diagonal entries are all equal to 1.
(b) Find a unipotent matrix which is neither lower nor upper triangular.

Chris Trentman
Chris Trentman
Numerade Educator
01:42

Problem 13

Write down all $4 \times 4$ permutation matrices that (a) fix the third row of a $4 \times 4$ matrix $A ;(b)$ take the third row to the fourth row; (c) interchange the second and third rows.

Nez Nikoo
Nez Nikoo
Numerade Educator
01:08

Problem 13

Prove that if $c \neq 0$ is any nonzero scalar and $A$ is an invertible matrix, then the scalar product matrix $c A$ is invertible, and $(c A)^{-1}=\frac{1}{c} A^{-1}$.

Chandra Jain
Chandra Jain
Numerade Educator
01:12

Problem 13

Let $A$ and $B$ be $m \times n$ matrices. (a) Suppose that $\mathbf{v}^T A \mathbf{w}=\mathbf{v}^T B \mathbf{w}$ for all vectors $\mathbf{v}, \mathbf{w}$. Prove that $A=B$. (b) Give an example of two matrices such that $\mathbf{v}^T A \mathbf{v}=\mathbf{v}^T B \mathbf{v}$ for all vectors $\mathbf{v}$, but $A \neq B$.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
08:05

Problem 13

Find the $L U$ factorizations of $\left(\begin{array}{lll}4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4\end{array}\right),\left(\begin{array}{llll}4 & 1 & 0 & 1 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 1 \\ 1 & 0 & 1 & 4\end{array}\right),\left(\begin{array}{lllll}4 & 1 & 0 & 0 & 1 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 \\ 1 & 0 & 0 & 1 & 4\end{array}\right)$.
Do you see a pattern? Try the $6 \times 6$ version. The following exercise should now be clear.

Nick Johnson
Nick Johnson
Numerade Educator
15:49

Problem 13

True or false: One can find an $m \times n$ matrix of rank $r$ for every $0 \leq r \leq \min \{m, n\}$.

Donald Albin
Donald Albin
Numerade Educator
00:45

Problem 13

Write out the formula for a $4 \times 4$ determinant. It should contain $24=4$ ! terms.

AG
Ankit Gupta
Numerade Educator
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Problem 14

Find all matrices $B$ that commute (under matrix multiplication) with $A=\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator
04:00

Problem 14

A square matrix $P$ is called idempotent if $P^2=P$. (a) Find all $2 \times 2$ idempotent upper triangular matrices. $(b)$ Find all $2 \times 2$ idempotent matrices.

Willis James
Willis James
Numerade Educator
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Problem 14

True or false: (a) Every elementary permutation matrix satisfies $P^2=I$. (b) Every permutation matrix satisfies $P^2=\mathrm{I}$. (c) A matrix that satisfies $P^2=\mathrm{I}$ is necessarily a permutation matrix.

Nick Johnson
Nick Johnson
Numerade Educator
04:13

Problem 14

Show that $A=\left(\begin{array}{lllll}0 & a & 0 & 0 & 0 \\ b & 0 & c & 0 & 0 \\ 0 & d & 0 & e & 0 \\ 0 & 0 & f & 0 & g \\ 0 & 0 & 0 & h & 0\end{array}\right)$ is not invertible for any value of the entries.

Victoria Chayes
Victoria Chayes
Numerade Educator
06:34

Problem 14

(a) Explain why the inverse of a permutation matrix equals its transpose: $P^{-1}=P^T$. (b) If $A^{-1}=A^T$, is $A$ necessarily a permutation matrix?

Victor Salazar
Victor Salazar
Numerade Educator
01:58

Problem 14

A tricirculant matrix $C=\left(\begin{array}{cccccc}q_1 & r_1 & & & & p_1 \\ p_2 & q_2 & r_2 & & & \\ & p_3 & q_3 & r_3 & & \\ & & \ddots & \ddots & \ddots & \\ & & & p_{n-1} & q_{n-1} & r_{n-1}\end{array}\right)$ is tridiagonal except for its $(1, n)$ and $(n, 1)$ entries. Tricirculant matrices arise in the numerical solution of periodic boundary value problems and in spline interpolation.
(a) Prove that if $C=L U$ is regular, its factors have the form
$$
\left(\begin{array}{ccccccccc}
1 & & & & & & \\
l_1 & 1 & & & & & \\
& l_2 & 1 & & & & \\
& & l_3 & 1 & & & \\
& & & \ddots & \ddots & & \\
& & & & l_{n-2} & 1 & \\
m_1 & m_2 & m_3 & \ldots & m_{n-2} & l_{n-1} & 1
\end{array}\right), \quad\left(\begin{array}{cccccccc}
d_1 & u_1 & & & & & & v_1 \\
& d_2 & u_2 & & & & v_2 \\
& & d_3 & u_3 & & & \\
& & \ddots & \ddots & & \\
& & & & v_{3-2} & u_{n-2} & v_{n-2} \\
& & & & & d_{n-1} & u_{n-1} \\
& & & & & & d_n
\end{array}\right) .
$$
(b) Compute the $L U$ factorization of the $n \times n$ tricirculant matrix
$$
C_n=\left(\begin{array}{rrrrrr}
1 & -1 & & & & -1 \\
-1 & 2 & -1 & & & \\
& -1 & 3 & -1 & & \\
& & \ddots & \ddots & \ddots & \\
& & & -1 & n-1 & -1 \\
-1 & & & & -1 & 1
\end{array}\right) \text { for } n=3,5 \text {, and } 6 \text {. What goes wrong when } n=4 \text { ? }
$$

Victor Salazar
Victor Salazar
Numerade Educator
01:03

Problem 14

True or false: Every $m \times n$ matrix has (a) exactly $m$ pivots; (b) at least one pivot.

Victor Salazar
Victor Salazar
Numerade Educator
02:14

Problem 14

Show that (1.87) satisfies all four determinant axioms, and hence is the correct formula for a determinant.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:52

Problem 15

(a) Show that, if $A, B$ are commuting square matrices, then $(A+B)^2=A^2+2 A B+B^2$.
(b) Find a pair of $2 \times 2$ matrices $A, B$ such that $(A+B)^2 \neq A^2+2 A B+B^2$.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
01:25

Problem 15

What elementary row operations do the following matrices represent? What size matrices do they apply to?
(a) $\left(\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 0 \\ 7 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & -5 \\ 0 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$.

WM
William Mead
Numerade Educator
01:54

Problem 15

(a) Let $P$ and $Q$ be $n \times n$ permutation matrices and $\mathbf{v} \in \mathbb{R}^n$ a vector. Under what conditions does the equation $P \mathbf{v}=Q \mathbf{v}$ imply that $P=Q$ ? (b) Answer the same question when $P A=Q A$, where $A$ is an $n \times k$ matrix.

Darsh Patel
Darsh Patel
Numerade Educator
02:45

Problem 15

Show that if $A$ is a nonsingular matrix, so is everyy power $A^n$.

Prashant Bana
Prashant Bana
Numerade Educator
06:09

Problem 15

Let $A$ be a square matrix and $P$ a permutation matrix of the same size. (a) Explain why the product $A P^T$ has the effect of applying the permutation defined by $P$ to the columns of $A$. (b) Explain the effect of multiplying $P A P^T$. Hint: Try this on some $3 \times 3$ examples first.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 15

A matrix $A$ is said to have bandwidth $k$ if all entries that are more than $k$ slots away from the main diagonal are zero: $a_{i j}=0$ whenever $|i-j|>k$. (a) Show that a tridiagonal matrix has band width 1 . (b) Write down an example of a $6 \times 6$ matrix of band width 2 and one of band width 3. (c) Prove that the $L$ and $U$ factors of a regular banded matrix have the same band width. (d) Find the $L U$ factorization of the matrices you wrote down in part (b). (e) Use the factorization to solve the system $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector with all entries equal to 1 . (f) How many arithmetic operations are needed to solve $A \mathbf{x}=\mathbf{b}$ if $A$ is banded? ( $g$ ) Prove or give a counterexample: the inverse of a banded matrix is banded.

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Problem 15

(a) Prove that the product $A=\mathbf{v} \mathbf{w}^T$ of a nonzero $m \times 1$ column vector $\mathbf{v}$ and a nonzero $1 \times n$ row vector $\mathbf{w}^T$ is an $m \times n$ matrix of rank $r=1$. (b) Compute the following rank one products: (i) $\left(\begin{array}{l}1 \\ 3\end{array}\right)\left(\begin{array}{ll}-1 & 2\end{array}\right)$,
(ii) $\left(\begin{array}{r}4 \\ 0 \\ -2\end{array}\right)\left(\begin{array}{ll}-2 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{r}2 \\ -3\end{array}\right)\left(\begin{array}{lll}1 & 3 & -1\end{array}\right)$.
(c) Prove that every rank one matrix can be written in the form $A=\mathbf{v} \mathbf{w}^T$.

Nick Johnson
Nick Johnson
Numerade Educator
02:51

Problem 15

Prove that axiom (iv) in Theorem 1.50 can be proved as a consequence of the first three axioms and the property det $\mathrm{I}=1$.

Brandon Collins
Brandon Collins
Numerade Educator
01:52

Problem 16

Show that if the matrices $A$ and $B$ commute, then they necessarily are both square and the same size.

WM
William Mead
Numerade Educator
View

Problem 16

Write down the elementary matrix corresponding to the following row operations on $4 \times 4$ matrices: (a) Add the third row to the fourth row. (b) Subtract the fourth row from the third row. (c) Add 3 times the last row to the first row. (d) Subtract twice the second row from the fourth row.

Victor Salazar
Victor Salazar
Numerade Educator
00:36

Problem 16

Let $P$ be the $3 \times 3$ permutation matrix such that the product $P A$ permutes the first and third rows of the $3 \times 3$ matrix $A$. (a) Write down $P$. (b) True or false: The product $A P$ is obtained by permuting the first and third columns of $A$.
(c) Does the same conclusion hold for every permutation matrix: is the effect of $P A$ on the rows of a square matrix $A$ the same as the effect of $A P$ on the columns of $A$ ?

Thomas Emment
Thomas Emment
Numerade Educator
01:08

Problem 16

Prove that a diagonal matrix $D=\operatorname{diag}\left(d_1, \ldots, d_n\right)$ is invertible if and only if all its diagonal entries are nonzero, in which case $D^{-1}=\operatorname{diag}\left(1 / d_1, \ldots, 1 / d_n\right)$.

Jacob Denson
Jacob Denson
Numerade Educator

Problem 16

Let $\mathbf{v}, \mathbf{w}$ be $n \times 1$ column vectors. (a) Prove that in most cases the inverse of the $n \times n$ matrix $A=\mathrm{I}-\mathbf{v} \mathbf{w}^T$ has the form $A^{-1}=\mathrm{I}-c \mathbf{v} \mathbf{w}^T$ for some scalar $c$. Find all $\mathbf{v}, \mathbf{w}$ for which such a result is valid. (b) Illustrate the method when $\mathbf{v}=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{r}-1 \\ 2\end{array}\right)$. (c) What happens when the method fails?

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01:21

Problem 16

(a) Find the exact solution to the linear system $\left(\begin{array}{rr}.1 & 2.7 \\ 1.0 & .5\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{r}10 . \\ -6.0\end{array}\right)$.
(b) Solve
the system using Gaussian Elimination with 2-digit rounding. (c) Solve the system using Partial Pivoting and 2-digit rounding. (d) Compare your answers and discuss.

James Kiss
James Kiss
Numerade Educator
06:35

Problem 16

(a) Let $A$ be an $m \times n$ matrix and let $M=(A \mid \mathbf{b})$ be the augmented matrix for the linear system $A \mathbf{x}=\mathbf{b}$. Show that either (i) $\operatorname{rank} A=\operatorname{rank} M$, or (ii) $\operatorname{rank} A=\operatorname{rank} M-1$.
(b) Prove that the system is compatible if and only if case (i) holds.

Uma Kumari
Uma Kumari
Numerade Educator
03:33

Problem 16

Prove that one cannot produce an elementary row operation of type \#2 by a combination of elementary row operations of type \#1.

Victoria Chayes
Victoria Chayes
Numerade Educator

Problem 17

Let $A$ be an $m \times n$ matrix. What are the permissible sizes for the zero matrices appearing in the identities $A \mathrm{O}=\mathrm{O}$ and $\mathrm{O} A=\mathrm{O}$ ?

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01:42

Problem 17

Compute the product $L_3 L_2 L_1$ of the elementary matrices (1.21). Compare your answer with (1.23).

Scott Stetson
Scott Stetson
Numerade Educator

Problem 17

A common notation for a permutation $\pi$ of the integers $\{1, \ldots, m\}$ is as a $2 \times m$ $\operatorname{matrix}\left(\begin{array}{ccccc}1 & 2 & 3 & \ldots & m \\ \pi(1) & \pi(2) & \pi(3) & \ldots & \pi(m)\end{array}\right)$, indicating that $\pi$ takes $i$ to $\pi(i)$. (a) Show that such a permutation corresponds to the permutation matrix with 1's in positions $(\pi(j), j)$ for $j=1, \ldots, m$. (b) Write down the permutation matrices corresponding to the following permutations:
(i) $\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3\end{array}\right)$,
(ii) $\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 2 & 3 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 1 & 4 & 2 & 3\end{array}\right)$, (iv) $\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1\end{array}\right)$. Which are elementary matrices? (c) Write down, using the preceding notation, the permutations corresponding to the following permutation matrices:
(i) $\left(\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right)$,
(ii) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$,
(iii) $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$,
(iv) $\left(\begin{array}{lllll}0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0\end{array}\right)$.

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05:24

Problem 17

Prove that if $U$ is a nonsingular upper triangular matrix, then the diagonal entries of $U^{-1}$ are the reciprocals of the diagonal entries of $U$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:36

Problem 17

Find all values of $a, b$, and $c$ for which the following matrices are symmetric:
(a) $\left(\begin{array}{cc}3 & a \\ 2 a-1 & a-2\end{array}\right)$,
(b) $\left(\begin{array}{ccc}1 & a & 2 \\ -1 & b & c \\ b & 3 & 0\end{array}\right)$,
(c)
$$
\left(\begin{array}{ccc}
3 & a+2 b-2 c & -4 \\
6 & 7 & b-c \\
-a+b+c & 4 & b+3 c
\end{array}\right) .
$$

AG
Ankit Gupta
Numerade Educator
01:58

Problem 17

(a) Find the exact solution to the linear system $x-5 y-z=1, \frac{1}{6} x-\frac{5}{6} y+z=0$, $2 x-y=3$. (b) Solve the system using Gaussian Elimination with 4-digit rounding.
(c) Solve the system using Partial Pivoting and 4-digit rounding. Compare your answers.

Adam Dehollander
Adam Dehollander
Numerade Educator
04:41

Problem 17

Find the rank of the matrix
$$
\begin{gathered}
\left(\begin{array}{cccc}
a & a r & \ldots & a r^{n-1} \\
a r^n & a r^{n+1} & \ldots & a r^{2 n-1} \\
\vdots & \vdots & \ddots & \vdots \\
a r^{(n-1) n} & a r^{(n-1) n+1} & \ldots & a r^{n^2-1}
\end{array}\right) \text { when } a, r \neq 0 . \\
\operatorname{matrix}\left(\begin{array}{ccccc}
1 & 2 & 3 & \ldots & \\
n+1 & n+2 & n+3 & \ldots & 2 n \\
2 n+1 & 2 n+2 & 2 n+3 & \ldots & 3 n \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
n^2-n+1 & n^2-n+2 & \ldots & \ldots & n^2
\end{array}\right) .
\end{gathered}
$$

Uma Kumari
Uma Kumari
Numerade Educator
01:57

Problem 17

Show that (a) if $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ is regular, then its pivots are $a$ and $\frac{\operatorname{det} A}{a}$;
(b) if $A=\left(\begin{array}{lll}a & b & e \\ c & d & f \\ g & h & j\end{array}\right)$ is regular, then its pivots are $a, \frac{a d-b c}{a}$, and $\frac{\operatorname{det} A}{a d-b c}$.
(c) Can you generalize this observation to regular $n \times n$ matrices?

Nick Derr
Nick Derr
Numerade Educator
02:46

Problem 18

Let $A$ be an $m \times n$ matrix and let $c$ be a scalar. Show that if $c A=\mathrm{O}$, then either $c=0$ or $A=\mathrm{O}$.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
09:15

Problem 18

Determine the product $E_3 E_2 E_1$ of the elementary matrices in (1.19). Is this the same as the product $E_1 E_2 E_3$ ? Which is easier to predict?

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator

Problem 18

Justify the statement that there are $n$ ! different $n \times n$ permutation matrices.

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01:16

Problem 18

(a) Let $U$ be a $m \times n$ matrix and $V$ an $n \times m$ matrix, such that the $m \times m$ matrix $\mathrm{I}_m+U V$ is invertible. Prove that $\mathrm{I}_n+V U$ is also invertible, and is given by
$$
\left(\mathrm{I}_n+V U\right)^{-1}=\mathrm{I}_n-V\left(\mathrm{I}_m+U V\right)^{-1} U \text {. }
$$
(b) The Sherman-Morrison-Woodbury formula generalizes this identity to
$$
(A+V B U)^{-1}=A^{-1}-A^{-1} V\left(B^{-1}+U A^{-1} V\right)^{-1} U A^{-1} .
$$
Explain what assumptions must be made on the matrices $A, B, U, V$ for (1.42) to be valid.

Sikandar Baig
Sikandar Baig
Numerade Educator
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Problem 18

List all symmetric (a) $3 \times 3$ permutation matrices, (b) $4 \times 4$ permutation matrices.

Victor Salazar
Victor Salazar
Numerade Educator
04:11

Problem 18

Answer Exercise 1.7.17 for the system
$$
x+4 y-3 z=-3, \quad 25 x+97 y-35 z=39, \quad 35 x-22 y+33 z=-15 .
$$

Nina Brocavich
Nina Brocavich
Numerade Educator
04:41

Problem 18

Find the rank of the $n \times n$ matrix

Uma Kumari
Uma Kumari
Numerade Educator
03:05

Problem 18

In this exercise, we justify the use of "elementary column operations" to compute determinants. Prove that (a) adding a scalar multiple of one column to another does not change the determinant; (b) multiplying a column by a scalar multiplies the determinant by the same scalar; $(c)$ interchanging two columns changes the sign of the determinant. (d) Explain how to use elementary column operations to reduce a matrix to lower triangular form and thereby compute its determinant.

AG
Ankit Gupta
Numerade Educator
01:05

Problem 19

True or false: If $A B=\mathrm{O}$ then either $A=\mathrm{O}$ or $B=\mathrm{O}$.

Carson Merrill
Carson Merrill
Numerade Educator
01:52

Problem 19

(a) Explain, using their interpretation as elementary row operations, why elementary matrices do not generally commute: $E \bar{E} \neq \bar{E} E$. (b) Which pairs of the elementary matrices listed in (1.19) commute? (c) Can you formulate a general rule that tells in advance whether two given elementary matrices commute?

WM
William Mead
Numerade Educator
04:38

Problem 19

Consider the following combination of elementary row operations of type \#1: (i) Add row $i$ to row $j$. (ii) Subtract row $j$ from row $i$. (iii) Add row $i$ to row $j$ again. Prove that the net effect is to interchange -1 times row $i$ with row $j$. Thus, we can almost produce an elementary row operation of type \#2 by a combination of elementary row operations of type \#1. Lest you be tempted to try, Exercise 1.9.16 proves that one cannot produce a bona fide row interchange by a combination of elementary row operations of type \#1.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:26

Problem 19

Two matrices $A$ and $B$ are said to be similar, written $A \sim B$, if there exists an invertible matrix $S$ such that $B=S^{-1} A S$. Prove: (a) $A \sim A$. (b) If $A \sim B$, then $B \sim A$. (c) If $A \sim B$ and $B \sim C$, then $A \sim C$.

Chandra Jain
Chandra Jain
Numerade Educator
View

Problem 19

True or false: If $A$ is symmetric, then $A^2$ is symmetric.

Nick Johnson
Nick Johnson
Numerade Educator
11:13

Problem 19

Employ 2 digit arithmetic with rounding to compute an approximate solution of the linear system $0.2 x+2 y-3 z=6,5 x+43 y+27 z=58,3 x+23 y-42 z=-87$, using the following methods: (a) Regular Gaussian Elimination with Back Substitution; (b) Gaussian Elimination with Partial Pivoting; (c) Gaussian Elimination with Full Pivoting. (d) Compare your answers and discuss their accuracy.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
01:04

Problem 19

Find two matrices $A, B$ such that $\operatorname{rank} A B \neq \operatorname{rank} B A$.

AG
Ankit Gupta
Numerade Educator
01:46

Problem 19

Find the determinant of the Vandermonde matrices listed in Exercise 1.3.24. Can you guess the general $n \times n$ formula?

Nick Johnson
Nick Johnson
Numerade Educator
03:33

Problem 20

True or false: If $A, B$ are square matrices of the same size, then
$$
A^2-B^2=(A+B)(A-B) \text {. }
$$

Patrick Burns
Patrick Burns
Numerade Educator
02:48

Problem 20

Determine which of the following $3 \times 3$ matrices is (i) upper triangular, (ii) upper unitriangular, (iii) lower triangular, and/or (iv) lower unitriangular:
(a) $\left(\begin{array}{rrr}1 & 2 & 0 \\ 0 & 3 & 2 \\ 0 & 0 & -2\end{array}\right)$
(b) $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$
(c) $\left(\begin{array}{lll}1 & 0 & 0 \\ 2 & 0 & 0 \\ 0 & 3 & 3\end{array}\right)$
(d) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & -4 & 1\end{array}\right)$
(e) $\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 0\end{array}\right)$.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:05

Problem 20

What is the effect of permuting the columns of its coefficient matrix on a linear system?

Neel Faucher
Neel Faucher
Numerade Educator
03:04

Problem 20

(a) A block matrix $D=\left(\begin{array}{ll}A & O \\ O & B\end{array}\right)$ is called block diagonal if $A$ and $B$ are square matrices, not necessarily of the same size, while the O's are zero matrices of the appropriate sizes. Prove that $D$ has an inverse if and only if both $A$ and $B$ do, and
$D^{-1}=\left(\begin{array}{cc}A^{-1} & \mathrm{O} \\ \mathrm{O} & B^{-1}\end{array}\right)$.
(b) Find the inverse of $\left(\begin{array}{lll}1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 3\end{array}\right)$ and $\left(\begin{array}{rrrr}1 & -1 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 2 & 5\end{array}\right)$ by using this method.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
View

Problem 20

True or false: If $A$ is a nonsingular symmetric matrix, then $A^{-1}$ is also symmetric.

Nick Johnson
Nick Johnson
Numerade Educator
00:43

Problem 20

Solve the following systems by hand, using pointers instead of physically interchanging
the rows:
(a) $\left(\begin{array}{rrr}0 & 1 & -2 \\ 1 & -1 & 1 \\ 3 & 1 & 0\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right)$,
(b)
$\left(\begin{array}{rrrr}0 & -1 & 0 & -1 \\ 0 & 0 & -2 & 1 \\ 1 & 0 & 2 & 0 \\ -1 & 0 & 0 & 3\end{array}\right)\left(\begin{array}{c}x \\ y \\ z \\ w\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right)$
(c) $\left(\begin{array}{rrrr}3 & -1 & 2 & -1 \\ 6 & -2 & 4 & 3 \\ 3 & 1 & 0 & -2 \\ -1 & 3 & -2 & 0\end{array}\right)\left(\begin{array}{c}x \\ y \\ z \\ w\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 1 \\ 1\end{array}\right)$,
(d)
$$
\left(\begin{array}{rrrr}
0 & -1 & 5 & -1 \\
1 & -2 & 0 & 1 \\
2 & -3 & -3 & -1 \\
2 & 0 & 1 & -1
\end{array}\right)\left(\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right)=\left(\begin{array}{r}
1 \\
-2 \\
3 \\
0
\end{array}\right)
$$

James Kiss
James Kiss
Numerade Educator

Problem 20

Let $A$ be an $m \times n$ matrix of rank $r$. (a) Suppose $C=(A B)$ is an $m \times k$ matrix, $k>n$, whose first $n$ columns are the same as the columns of $A$. Prove that $\operatorname{rank} C \geq \operatorname{rank} A$. Give an example with $\operatorname{rank} C=\operatorname{rank} A$; with $\operatorname{rank} C>\operatorname{rank} A$. (b) Let $E=\left(\begin{array}{c}A \\ D\end{array}\right)$ be a $j \times n$ matrix, $j>m$, whose first $m$ rows are the same as those of $A$. Prove that $\operatorname{rank} E \geq \operatorname{rank} A$. Give an example with $\operatorname{rank} E=\operatorname{rank} A$; with $\operatorname{rank} E>\operatorname{rank} A$.

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Problem 20

Cramer's Rule. (a) Show that the nonsingular system $a x+b y=p, c x+d y=q$ has the solution given by the determinantal ratios
$$
x=\frac{1}{\Delta} \operatorname{det}\left(\begin{array}{ll}
p & b \\
q & d
\end{array}\right), \quad y=\frac{1}{\Delta} \operatorname{det}\left(\begin{array}{ll}
a & p \\
c & q
\end{array}\right), \quad \text { where } \quad \Delta=\operatorname{det}\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) .
$$
(b) Use Cramer's Rule (1.90) to solve the systems
(i)
$$
\begin{aligned}
& x+3 y=13, \\
& 4 x+2 y=0 \text {, } \\
& \text { (ii) } 3 x+6 y=-2 \text {. } \\
&
\end{aligned}
$$
(ii) $x-2 y=4$,
$$
a x+b y+c z=p,
$$
(c) Prove that the solution to
$$
d x+e y+f z=q
$$
$$
\text { (2i) } 3 x+6 y=-2 \text {. }
$$
$$
\begin{aligned}
& d x+e y+f z=q, \\
& g x+h y+j z=r,
\end{aligned}
$$
with $\Delta=\operatorname{det}\left(\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & j\end{array}\right) \neq 0$ is
$$
\begin{aligned}
& x=\frac{1}{\Delta} \operatorname{det}\left(\begin{array}{lll}
p & b & c \\
q & e & f \\
r & h & j
\end{array}\right), \quad y=\frac{1}{\Delta} \operatorname{det}\left(\begin{array}{lll}
a & p & c \\
d & q & f \\
g & r & j
\end{array}\right), \quad z=\frac{1}{\Delta} \operatorname{det}\left(\begin{array}{ccc}
a & b & p \\
d & e & q \\
g & h & r
\end{array}\right) \text {. } \\
& x+4 y=3, \quad 3 x+2 y-z=1, \\
&
\end{aligned}
$$
(d) Use Cramer's Rule (1.91) to solve
(i)
$$
\begin{aligned}
& 4 x+2 y+z=2 \text {, } \\
& x-3 y+2 z=2 \text {, } \\
& -x+y-z=0, \\
& 2 x-y+z=3 \text {. } \\
&
\end{aligned}
$$
(ii)
(e) Can you see the pattern that will generalize to $n$ equations in $n$ unknowns?
Remark. Although elegant, Cramer's rule is not a very practical solution method.

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04:04

Problem 21

Prove that $A \mathbf{v}=\mathbf{0}$ for every vector $\mathbf{v}$ (with the appropriate number of entries) if and only if $A=\mathrm{O}$ is the zero matrix. Hint: If you are stuck, first try to find a proof when $A$ is a small matrix, e.g., of size $2 \times 2$.

Lucas Finney
Lucas Finney
Numerade Educator
View

Problem 21

Find the $L U$ factorization of the following matrices:
(a) $\left(\begin{array}{rr}1 & 3 \\ -1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}-1 & 1 & -1 \\ 1 & 1 & 1 \\ -1 & 1 & 2\end{array}\right)$,
(d) $\left(\begin{array}{lll}2 & 0 & 3 \\ 1 & 3 & 1 \\ 0 & 1 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}-1 & 0 & 0 \\ 2 & -3 & 0 \\ 1 & 3 & 2\end{array}\right)$,
(f) $\left(\begin{array}{rrr}1 & 0 & -1 \\ 2 & 3 & 2 \\ -3 & 1 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 2 & -1 & -1 \\ -1 & 3 & 0 & 2 \\ 0 & -1 & 2 & 1\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}1 & 1 & -2 & 3 \\ -1 & 2 & 3 & 0 \\ -2 & 1 & 1 & -2 \\ 3 & 0 & 1 & 5\end{array}\right)$,
(i) $\left(\begin{array}{llll}2 & 1 & 3 & 1 \\ 1 & 4 & 0 & 1 \\ 3 & 0 & 2 & 2 \\ 1 & 1 & 2 & 2\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
04:54

Problem 21

For each of the listed matrices $A$ and vectors $\mathbf{b}$, find a permuted $L U$ factorization of the matrix, and use your factorization to solve the system $A \mathbf{x}=\mathbf{b}$. (a) $\left(\begin{array}{lr}0 & 1 \\ 2 & -1\end{array}\right),\left(\begin{array}{l}3 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{rrr}0 & 0 & -4 \\ 1 & 2 & 3 \\ 0 & 1 & 7\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & 1 & -3 \\ 0 & 2 & 3 \\ 1 & 0 & 2\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$,
(d) $\left(\begin{array}{rrrr}1 & 2 & -1 & 0 \\ 3 & 6 & 2 & -1 \\ 1 & 1 & -7 & 2 \\ 1 & -1 & 2 & 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 3\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}0 & 1 & 0 & 0 \\ 2 & 3 & 1 & 0 \\ 1 & 4 & -1 & 2 \\ 7 & -1 & 2 & 3\end{array}\right),\left(\begin{array}{r}-1 \\ -4 \\ 0 \\ 5\end{array}\right)$,
(f) $\left(\begin{array}{rrrrr}0 & 0 & 2 & 3 & 4 \\ 0 & 1 & -7 & 2 & 3 \\ 1 & 4 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 7 & 3\end{array}\right),\left(\begin{array}{r}-3 \\ -2 \\ 0 \\ 0 \\ -7\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator
03:33

Problem 21

(a) Show that $B=\left(\begin{array}{rrr}1 & 1 & 0 \\ -1 & -1 & 1\end{array}\right)$ is a left inverse of $A=\left(\begin{array}{rr}1 & -1 \\ 0 & 1 \\ 1 & 1\end{array}\right)$. (b) Show that $A$ does not have a right inverse. (c) Can you find any other left inverses of $A$ ?

David Mccaslin
David Mccaslin
Numerade Educator
03:33

Problem 21

True or false: If $A$ and $B$ are symmetric $n \times n$ matrices, so is $A B$.

Patrick Burns
Patrick Burns
Numerade Educator
00:46

Problem 21

Solve the following systems using Partial Pivoting and pointers:
(a)
$$
\begin{aligned}
& \text { (a) }\left(\begin{array}{rr}
1 & 5 \\
2 & -3
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)=\left(\begin{array}{r}
3 \\
-2
\end{array}\right), \quad \text { (b) }\left(\begin{array}{rrr}
1 & 2 & -1 \\
4 & -2 & 1 \\
3 & 5 & -1
\end{array}\right)\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)=\left(\begin{array}{l}
1 \\
3 \\
1
\end{array}\right), \\
& \text { (c) }\left(\begin{array}{rrrr}
1 & -3 & 6 & -1 \\
2 & -5 & 0 & 1 \\
-1 & -6 & 4 & -2 \\
3 & 0 & 2 & -1
\end{array}\right)\left(\begin{array}{l}
x \\
y \\
z \\
w
\end{array}\right)=\left(\begin{array}{r}
1 \\
-2 \\
0 \\
1
\end{array}\right), \quad(d)\left(\begin{array}{rrr}
01 & 4 & 2 \\
2 & -802 & 3 \\
7 & .03 & 250
\end{array}\right)\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)=\left(\begin{array}{r}
1 \\
2 \\
122
\end{array}\right) .
\end{aligned}
$$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:16

Problem 21

Let $A$ be a singular square matrix. Prove that there exist elementary matrices $E_1, \ldots, E_N$ such that $A=E_1 E_2 \cdots E_N Z$, where $Z$ is a matrix with at least one all-zero row.

Vaibhav Jain
Vaibhav Jain
Numerade Educator
08:58

Problem 21

(a) Show that if $D=\left(\begin{array}{ll}A & O \\ O & B\end{array}\right)$ is a block diagonal matrix, where $A$ and $B$ are square matrices, then $\operatorname{det} D=\operatorname{det} A \operatorname{det} B$. (b) Prove that the same holds for a block upper triangular matrix $\operatorname{det}\left(\begin{array}{ll}A & C \\ O & B\end{array}\right)=\operatorname{det} A \operatorname{det} B . \quad(c)$ Use this method to compute the determinant of the following matrices:
(i) $\left(\begin{array}{rrr}3 & 2 & -2 \\ 0 & 4 & -5 \\ 0 & 3 & 7\end{array}\right)$,
(ii) $\left(\begin{array}{rrrr}1 & 2 & -2 & 5 \\ -3 & 1 & 0 & -5 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 2 & -2\end{array}\right)$,
(iii) $\left(\begin{array}{rrrr}1 & 2 & 0 & 4 \\ -3 & 1 & 4 & -1 \\ 0 & 3 & 1 & 8 \\ 0 & 0 & 0 & -3\end{array}\right)$,
(iv) $\left(\begin{array}{rrrr}5 & -1 & 0 & 0 \\ 2 & 5 & 0 & 0 \\ 2 & 4 & 4 & -2 \\ 3 & -2 & 9 & -5\end{array}\right)$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 22

(a) Under what conditions is the square $A^2$ of a matrix defined? (b) Show that $A$ and $A^2$ commute. (c) How many matrix multiplications are needed to compute $A^n$ ?

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13:38

Problem 22

Given the factorization $A=\left(\begin{array}{rrr}2 & -1 & 0 \\ -6 & 4 & -1 \\ 4 & -6 & 7\end{array}\right)=\left(\begin{array}{rrr}1 & 0 & 0 \\ -3 & 1 & 0 \\ 2 & -4 & 1\end{array}\right)\left(\begin{array}{rrr}2 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 3\end{array}\right)$, explain, without computing, which elementary row operations are used to reduce $A$ to upper triangular form. Be careful to state the order in which they should be applied. Then check the correctness of your answer by performing the elimination.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 22

For each of the following linear systems find a permuted $L U$ factorization of the coefficient matrix and then use it to solve the system by Forward and Back Substitution.
$$
\begin{aligned}
4 x_1-4 x_2+2 x_3 & =1, \\
-3 x_1+3 x_2+x_3 & =3, \\
-3 x_1+x_2-2 x_3 & =-5
\end{aligned}
$$
(a) $-3 x_1+3 x_2+x_3=3$,
$$
y-z+w=0,
$$
$$
y+z=1,
$$
(b)
$$
\begin{aligned}
& x-y+z-3 w=2, \\
& x+2 y-z+w=4 .
\end{aligned}
$$
$$
\begin{aligned}
x-y+2 z+w & =0, \\
-x+y-3 z & =1,
\end{aligned}
$$
(c)
$$
\begin{aligned}
x-y+4 z-3 w & =2, \\
x+2 y-z+w & =4 .
\end{aligned}
$$
0\end{array}\right)$.
Do you always obtain the same result? Explain why or why not.
1.4.24. (a) Find three different permuted $L U$ factorizations of the matrix $A=\left(\begin{array}{rrr}0 & 1 & 2 \\ 1 & 0 & -1 \\ 1 & 1 & 3\end{array}\right)$. (b) How many different permuted $L U$ factorizations does $A$ have?

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02:48

Problem 22

Prove that the rectangular matrix $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 1 & 2 & 0\end{array}\right)$ has a right inverse, but no left inverse.

Mauricio Araiza Canizales
Mauricio Araiza Canizales
Numerade Educator
01:02

Problem 22

(a) Show that every diagonal matrix is symmetric. (b) Show that an upper (lower) triangular matrix is symmetric if and only if it is diagonal.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 22

Use Full Pivoting with pointers to solve the systems in Exercise 1.7.21.

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Problem 22

Solve the following homogeneous linear systems.
(a)
$$
\begin{aligned}
x+y-2 z & =0, \\
-x+4 y-3 z & =0 .
\end{aligned}
$$
(d)
$$
\begin{aligned}
x+2 y-2 z+w & =0, \\
-3 x+z-2 w & =0 .
\end{aligned}
$$
(b)
$$
2 x+3 y-z=0,
$$
$$
-x+y-4 z=0,
$$
(c)
$$
x-3 y+3 z=0 \text {. }
$$
$$
-x+3 y-2 z+w=0,
$$
(e) $-2 x+5 y+z-2 w=0$,
$$
3 x-8 y+z-4 w=0 \text {. }
$$
$-2 x+2 y-6 z=0$,
$x+3 y+3 z=0$.
$$
-y+z=0,
$$
$$
2 x-3 w=0,
$$
(f)
$$
\begin{aligned}
& x+y-2 w=0, \\
& y-3 z+w=0 .
\end{aligned}
$$

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View

Problem 23

Find a nonzero matrix $A \neq \mathrm{O}$ such that $A^2=\mathrm{O}$.

Michelle Z.
Michelle Z.
Numerade Educator
00:56

Problem 23

(a) Write down a $4 \times 4$ lower unitriangular matrix whose entries below the diagonal are distinct nonzero numbers. (b) Explain which elementary row operation each entry corresponds to. (c) Indicate the order in which the elementary row operations should be performed by labeling the entries $1,2,3, \ldots$.

AG
Ankit Gupta
Numerade Educator
01:17

Problem 23

(a) Explain why
$$
\begin{aligned}
& \left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{rrr}
0 & 1 & 3 \\
2 & -1 & 1 \\
2 & -2 & 0
\end{array}\right)=\left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & -1 & 1
\end{array}\right)\left(\begin{array}{rrr}
2 & -1 & 1 \\
0 & 1 & 3 \\
0 & 0 & 2
\end{array}\right), \\
& \left(\begin{array}{lll}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{array}\right)\left(\begin{array}{rrr}
0 & 1 & 3 \\
2 & -1 & 1 \\
2 & -2 & 0
\end{array}\right)=\left(\begin{array}{lll}
1 & 0 & 0 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{array}\right)\left(\begin{array}{rrr}
2 & -2 & 0 \\
0 & 1 & 1 \\
0 & 0 & 2
\end{array}\right), \\
& \left(\begin{array}{lll}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right)\left(\begin{array}{rrr}
0 & 1 & 3 \\
2 & -1 & 1 \\
2 & -2 & 0
\end{array}\right)=\left(\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 1 & 1
\end{array}\right)\left(\begin{array}{rrr}
2 & -2 & 0 \\
0 & 1 & 3 \\
0 & 0 & -2
\end{array}\right), \\
&
\end{aligned}
$$
are all legitimate permuted $L U$ factorizations of the same matrix. List the elementary row operations that are being used in each case.
(b) Use each of the factorizations to solve the linear system $\left(\begin{array}{rrr}0 & 1 & 3 \\ 2 & -1 & 1 \\ 2 & -2 & 0\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}-5 \\ -1 \\

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:44

Problem 23

(a) Are there any $\mathrm{n}$
(b) Are there any nonsin

Liuxi Sun
Liuxi Sun
Numerade Educator
03:21

Problem 23

Let $A$ be a symmetric matrix. (a) Show that $A^n$ is symmetric for every nonnegative integer $n$. (b) Show that $2 A^2-3 A+\mathrm{I}$ is symmetric. (c) Show that every matrix polynomial $p(A)$ of $A$, cf. Exercise 1.2 .35 , is a symmetric matrix.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
View

Problem 23

Let $H_n$ be the $n \times n$ Hilbert matrix (1.72), and $K_n=H_n^{-1}$ its inverse. It can be proved, $[40 ;$ p. 513$]$, that the $(i, j)$ entry of $K_n$ is
$$
(-1)^{i+j}(i+j-1)\left(\begin{array}{c}
n+i-1 \\
n-j
\end{array}\right)\left(\begin{array}{c}
n+j-1 \\
n-i
\end{array}\right)\left(\begin{array}{c}
i+j-2 \\
i-1
\end{array}\right)^2 \text {, }
$$
where $\left(\begin{array}{l}n \\ k\end{array}\right)=\frac{n !}{k !(n-k) !}$ is the standard binomial coefficient. (Warning. Proving this formula is a nontrivial combinatorial challenge.) (a) Write down the inverse of the Hilbert matrices $\mathrm{H}_3, \mathrm{H}_4, \mathrm{H}_5$ using the formula or the Gauss-Jordan Method with exact rational arithmetic. Check your results by multiplying the matrix by its inverse.
(b) Recompute the inverses on your computer using floating point arithmetic and compare with the exact answers. (c) Try using floating point arithmetic to find $K_{10}$ and $K_{20}$. Test the answer by multiplying the Hilbert matrix by its computed inverse.

Nick Johnson
Nick Johnson
Numerade Educator
03:15

Problem 23

Find all solutions to the homogeneous system $A \mathbf{x}=\mathbf{0}$ for the coefficient matrix
(a) $\left(\begin{array}{rr}3 & -1 \\ -9 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & -1 & 4 \\ 3 & 1 & 2\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}1 & -2 & 3 & -3 \\ 2 & 1 & 4 & 0\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$,
(e) $\left(\begin{array}{rrr}0 & 2 & -1 \\ -2 & 0 & 3 \\ 1 & 3 & 0\end{array}\right)$,
(f) $\left(\begin{array}{rr}1 & -2 \\ 1 & -1 \\ 2 & -1 \\ 1 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & 2 & 0 \\ -1 & -3 & 2 \\ 4 & 7 & 2 \\ -1 & 1 & 6\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}0 & 0 & 3 & -3 \\ 1 & -1 & 0 & 3 \\ 2 & -2 & 1 & 5 \\ -1 & 1 & 1 & -4\end{array}\right)$.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:48

Problem 24

Let $A$ have a row all of whose entries are zero. (a) Explain why the product $A B$ also has a zero row. (b) Find an example where $B A$ does not have a zero row.

JC
Jeff Christopher
Numerade Educator
View

Problem 24

Let $t_1, t_2, \ldots$ be distinct real numbers. Find the $L U$ factorization of the following
Vandermonde matrices:
(a) $\left(\begin{array}{cc}1 & 1 \\ t_1 & t_2\end{array}\right)$,
(b) $\left(\begin{array}{ccc}1 & 1 & 1 \\ t_1 & t_2 & t_3 \\ t_1^2 & t_2^2 & t_3^2\end{array}\right)$,
(c) $\left(\begin{array}{cccc}1 & 1 & 1 & 1 \\ t_1 & t_2 & t_3 & t_4 \\ t_1^2 & t_2^2 & t_3^2 & t_4^2 \\ t_1^3 & t_2^3 & t_3^3 & t_4^3\end{array}\right)$.
Can you spot a pattern? Test your conjecture with the $5 \times 5$ Vandermonde matrix.

Victor Salazar
Victor Salazar
Numerade Educator
00:57

Problem 24

(a) Find three different permuted $L U$ factorizations of the matrix $A=\left(\begin{array}{rrr}0 & 1 & 2 \\ 1 & 0 & -1 \\ 1 & 1 & 3\end{array}\right)$.
(b) How many different permuted $L U$ factorizations does $A$ have?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:55

Problem 24

(a) Write down the elementary matrix that multiplies the third row of a $4 \times 4$ matrix by 7. (b) Write down its inverse.

Chandler Austin
Chandler Austin
Numerade Educator
02:40

Problem 24

Show that if $A$ is any matrix, then $K=A^T A$ and $L=A A^T$ are both well-defined, symmetric matrices.

Urvashi Arora
Urvashi Arora
Numerade Educator
05:59

Problem 24

(a) Write out a pseudo-code algorithm, using both row and column pointers, for Gaussian Elimination with Full Pivoting. (b) Implement your code on a computer, and try it on the systems in Exercise 1.7.21.

Jingyun Wang
Jingyun Wang
Numerade Educator
10:19

Problem 24

Let $U$ be an upper triangular matrix. Show that the homogeneous system $U \mathbf{x}=\mathbf{0}$ admits a nontrivial solution if and only if $U$ has at least one 0 on its diagonal.

RB
Rashi Bhatt
Numerade Educator
05:56

Problem 25

(a) Find all solutions $X=\left(\begin{array}{cc}x & y \\ z & w\end{array}\right)$ to the matrix equation $A X=B$ when $A=\left(\begin{array}{rr}0 & 1 \\ -1 & 3\end{array}\right)$ and $B=\left(\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right)$. (b) Find all solutions to $X A=B$. Are they the same?

Sam Stansfield
Sam Stansfield
Numerade Educator
02:48

Problem 25

Write down the explicit requirements on its entries $a_{i j}$ for a square matrix $A$ to be (a) diagonal, (b) upper triangular, (c) upper unitriangular, (d) lower triangular, (e) lower unitriangular.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:34

Problem 25

What is the maximal number of permuted $L U$ factorizations a regular $3 \times 3$ matrix can have? Give an example of such a matrix.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 25

Find the inverse of each of the following matrices, if possible, by applying the GaussJordan Method.
(a) $\left(\begin{array}{ll}1 & -2 \\ 3 & -3\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}\frac{3}{5} & -\frac{4}{5} \\ \frac{4}{5} & \frac{3}{5}\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & 0 & -2 \\ 3 & -1 & 0 \\ -2 & 1 & -3\end{array}\right)$,
(f) $\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 5 & 5 \\ 2 & 1 & 2\end{array}\right)$,
(g) $\left(\begin{array}{rrr}2 & 1 & 2 \\ 4 & 2 & 3 \\ 0 & -1 & 1\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}2 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & -2 & -5\end{array}\right)$,
(i) $\left(\begin{array}{rrrr}1 & -2 & 1 & 1 \\ 2 & -3 & 3 & 0 \\ 3 & -7 & 2 & 4 \\ 0 & 2 & 1 & 1\end{array}\right)$.
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1 Linear Algebraic Systems

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03:10

Problem 25

Find the $L D L^T$ factorization of the following symmetric matrices:
(a) $\left(\begin{array}{ll}1 & 1 \\ 1 & 4\end{array}\right)$,
(b) $\left(\begin{array}{rr}-2 & 3 \\ 3 & -1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & -1 & -1 \\ -1 & 3 & 2 \\ -1 & 2 & 0\end{array}\right)$,
(d)
$$
\left(\begin{array}{rrrr}
1 & -1 & 0 & 3 \\
-1 & 2 & 2 & 0 \\
0 & 2 & -1 & 0 \\
3 & 0 & 0 & 1
\end{array}\right)
$$

Victor Salazar
Victor Salazar
Numerade Educator
08:02

Problem 25

Find the solution to the homogeneous system $2 x_1+x_2-2 x_3=0,2 x_1-x_2-2 x_3=0$. Then solve the inhomogeneous version where the right-hand sides are changed to $a, b$, respectively. What do you observe?

Christian Otero
Christian Otero
Numerade Educator
05:56

Problem 26

(a) Find all solutions $X=\left(\begin{array}{cc}x & y \\ z & w\end{array}\right)$ to the matrix equation $A X=B$ when $A=\left(\begin{array}{rr}0 & 1 \\ -1 & 3\end{array}\right)$ and $B=\left(\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right)$.
(b) Find all solutions to $X A=B$. Are they the same?

Sam Stansfield
Sam Stansfield
Numerade Educator
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Problem 26

(a) Explain why the product of two lower triangular matrices is lower triangular. (b) What can you say concerning the diagonal entries of the product of two lower triangular matrices? (c) Explain why the product of two lower unitriangular matrices is also lower unitriangular.

Victor Salazar
Victor Salazar
Numerade Educator
00:34

Problem 26

True or false: The pivots of a nonsingular matrix are uniquely defined.

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
06:34

Problem 26

Write each of the matrices in Exercise 1.5.25 as a product of elementary matrices.

Isaac Chettiath
Isaac Chettiath
Numerade Educator
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Problem 26

Find the $L D L^T$ factorization of the matrices
$$
M_2=\left(\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right), \quad M_3=\left(\begin{array}{lll}
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{array}\right), \quad \text { and } \quad M_4=\left(\begin{array}{llll}
2 & 1 & 0 & 0 \\
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 2
\end{array}\right) .
$$

Victor Salazar
Victor Salazar
Numerade Educator
09:06

Problem 26

Answer Exercise 1.8.25 for the system $2 x_1+x_2+x_3-x_4=0,2 x_1-2 x_2-x_3+3 x_4=0$.

Jingyun Wang
Jingyun Wang
Numerade Educator
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Problem 27

(a) Find all solutions $X=\left(\begin{array}{cc}x & y \\ z & w\end{array}\right)$ to the matrix equation $A X=X B$ when $A=\left(\begin{array}{rr}1 & 2 \\ -1 & 0\end{array}\right)$ and $B=\left(\begin{array}{ll}0 & 1 \\ 3 & 0\end{array}\right)$. (b) Can you find a pair of nonzero matrices $A \neq B$ such that the matrix equation $A X=X B$ has a nonzero solution $X \neq \mathrm{O}$ ?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 27

(a) Find all solutions $X=\left(\begin{array}{cc}x & y \\ z & w\end{array}\right)$ to the matrix equation $A X=X B$ when $A=\left(\begin{array}{rr}1 & 2 \\ -1 & 0\end{array}\right)$ and $B=\left(\begin{array}{ll}0 & 1 \\ 3 & 0\end{array}\right)$.
(b) Can you find a pair of nonzero matrices $A \neq B$ such that the matrix equation $A X=X B$ has a nonzero solution $X \neq \mathrm{O}$ ?

Nick Johnson
Nick Johnson
Numerade Educator
01:17

Problem 27

True or false: If $A$ has a zero entry on its main diagonal, it is not regular.

Cheyenne Whinham
Cheyenne Whinham
Numerade Educator
02:06

Problem 27

(a) Write a pseudocode program implementing the algorithm for finding the permuted $L U$ factorization of a matrix. (b) Program your algorithm and test it on the examples in Exercise 1.4.21.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:06

Problem 27

Express $A=\left(\begin{array}{cc}\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right)$ as a product of elementary matrices.

Isaac Chettiath
Isaac Chettiath
Numerade Educator
05:26

Problem 27

Prove that the $3 \times 3$ matrix $A=\left(\begin{array}{rrr}1 & 2 & 1 \\ 2 & 4 & -1 \\ 1 & -1 & 3\end{array}\right)$ cannot be factored as $A=L D L^T$.

Wasim Sher
Wasim Sher
Numerade Educator
06:01

Problem 27

Find all values of $k$ for which the following homogeneous systems of linear equations have a non-trivial solution:
(a)
$$
\begin{aligned}
x+k y & =0, \\
k x+4 y & =0,
\end{aligned}
$$
$$
x_1+k x_2+4 x_3=0,
$$
(b)
$$
\begin{array}{r}
k x_1+x_2+2 x_3=0 \\
2 x_1+k x_2+8 x_3=0 .
\end{array}
$$
(c)
$$
\begin{aligned}
x+k y+2 z & =0, \\
3 x-k y-2 z & =0,
\end{aligned}
$$
$$
\begin{aligned}
(k+1) x-2 y-4 z & =0, \\
k x+3 y+6 z & =0 .
\end{aligned}
$$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:00

Problem 28

Let $A$ be a matrix and $c$ a scalar. Find all solutions to the matrix equation $c A=1$.

Sirat Shah
Sirat Shah
Numerade Educator
03:00

Problem 28

Let $A$ be a matrix and $c$ a scalar. Find all solutions to the matrix equation $c A=\mathrm{I}$.

Sirat Shah
Sirat Shah
Numerade Educator
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Problem 28

In general, how many elementary row operations does one need to perform in order to reduce a regular $n \times n$ matrix to upper triangular form?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 28

Use the Gauss-Jordan Method to find the inverse of the following complex matrices:
(a) $\left(\begin{array}{ll}\mathrm{i} & 1 \\ 1 & \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{cc}1 & 1-\mathrm{i} \\ 1+\mathrm{i} & 1\end{array}\right)$,
(c) $\left(\begin{array}{ccc}0 & 1 & -\mathrm{i} \\ \mathrm{i} & 0 & -1 \\ -1 & \mathrm{i} & 1\end{array}\right)$,
(d) $\left(\begin{array}{ccc}1 & 0 & \mathrm{i} \\ \mathrm{i} & -1 & 1+\mathrm{i} \\ -3 \mathrm{i} & 1-\mathrm{i} & 1+\mathrm{i}\end{array}\right)$.

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03:19

Problem 28

Skew-symmetric matrices: An $n \times n$ matrix $J$ is called skew-symmetric if $J^T=-J$. (a) Show that every diagonal entry of a skew-symmetric matrix is zero. (b) Write down an example of a nonsingular skew-symmetric matrix. (c) Can you find a regular skewsymmetric matrix? (d) Show that if $J$ is a nonsingular skew-symmetric matrix, then $J^{-1}$ is also skew-symmetric. Verify this fact for the matrix you wrote down in part (b). (e) Show that if $J$ and $K$ are skew-symmetric, then so are $J^T, J+K$, and $J-K$. What about $J K$ ? (f) Prove that if $J$ is a skew-symmetric matrix, then $\mathbf{v}^T J \mathbf{v}=0$ for all vectors $\mathbf{v} \in \mathbb{R}^n$.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
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Problem 29

Let $\mathrm{e}$ be the $1 \times m$ row vector all of whose entries are equal to 1 . (a) Show that if $A$ is an $m \times n$ matrix, then the $i^{\text {th }}$ entry of the product $\mathbf{v}=\mathbf{e} A$ is the $j^{\text {th }}$ column sum of $A$, meaning the sum of all the entries in its $j^{\text {th }}$ row. (b) Let $W$ denote the $m \times m$ matrix whose diagonal entries are equal to $\frac{1-m}{m}$ and whose off-diagonal entries are all equal to $\frac{1}{m}$. Prove that the column sums of $B=W A$ are all zero. (c) Check both results when $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 1 & 3 \\ -4 & 5 & -1\end{array}\right)$. Remark. If the rows of $A$ represent experimental data values, then the entries of $\frac{1}{m}$ e $A$ represent the means or averages of the data values, while $B=W A$ corresponds to data that has been normalized to have mean 0; see Section 8.8.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 29

Prove that if $A$ is a regular $2 \times 2$ matrix, then its $L U$ factorization is unique. In other words, if $A=L U=\widehat{L} \hat{U}$ where $L, \hat{L}$ are lower unitriangular and $U, \hat{U}$ are upper triangular, then $L=\hat{L}$ and $U=\hat{U}$. (The general case appears in Proposition 1.30.)

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01:30

Problem 29

Can two nonsingular linear systems have the same solution and yet not be equivalent?

Joseph Lentino
Joseph Lentino
Numerade Educator
04:54

Problem 29

(a) Prove that every square matrix can be expressed as the sum, $A=S+J$, of a symmetric matrix $S=S^T$ and a skew-symmetric matrix $J=-J^T$.
(b) Write $\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$ and $\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$ as the sum of symmetric and skew-symmetric matrices.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:32

Problem 30

The commutator of two matrices $A, B$, is defined to be the matrix
$$
C=[A, B]=A B-B A \text {. }
$$
(a) Explain why $[A, B]$ is defined if and only if $A$ and $B$ are square matrices of the same size. (b) Show that $A$ and $B$ commute under matrix multiplication if and only if $[A, B]=\mathrm{O}$. (c) Compute the commutator of the following matrices:
(i) $\left(\begin{array}{rr}1 & 0 \\ 1 & -1\end{array}\right),\left(\begin{array}{rr}2 & 1 \\ -2 & 0\end{array}\right)$;
(ii) $\left(\begin{array}{rr}-1 & 3 \\ 3 & -1\end{array}\right),\left(\begin{array}{ll}1 & 7 \\ 7 & 1\end{array}\right)$;
(iii) $\left(\begin{array}{rrr}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right),\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{array}\right)$;
(d) Prove that the commutator is (i) Bilinear: $[c A+d B, C]=c[A, C]+d[B, C]$ and $[A, c B+d C]=c[A, B]+d[A, C]$ for any scalars $c, d$; (ii) Skew-symmetric: $[A, B]=-[B, A] ;$ (iii) satisfies the the Jacobi identity:
$$
[[A, B], C]+[[C, A], B]+[[B, C], A]=\mathrm{O},
$$
for any square matrices $A, B, C$ of the same size.
Remark. The commutator plays a very important role in geometry, symmetry, and quantum mechanics. See Section 10.4 as well as $[\mathbf{5 4}, \mathbf{6 0}, \mathbf{9 3}]$ for further developments.

Raj Bala
Raj Bala
Numerade Educator
04:47

Problem 30

Prove directly that the matrix $A=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$ does not have an $L U$ factorization.

Mohan Jain
Mohan Jain
Numerade Educator
01:48

Problem 30

(a) Suppose $\bar{A}$ is obtained from $A$ by applying an elementary row operation. Let $C=A B$, where $B$ is any matrix of the appropriate size. Explain why $\bar{C}=\bar{A} B$ can be obtained by applying the same elementary row operation to $C$. (b) Illustrate by adding -2 times the first row to the third row of $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & -3 & 2 \\ 0 & 1 & -4\end{array}\right)$ and then multiplying the result on the right by $B=\left(\begin{array}{rr}1 & -2 \\ 3 & 0 \\ -1 & 1\end{array}\right)$. Check that the resulting matrix is the same as first multiplying $A B$ and then applying the same row operation to the product matrix.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 30

Suppose $A=L U$ is a regular matrix. Write down the $L U$ factorization of $A^T$. Prove that $A^T$ is also regular, and its pivots are the same as the pivots of $A$.

Victor Salazar
Victor Salazar
Numerade Educator
01:33

Problem 31

The trace of a $n \times n$ matrix $A \in \mathcal{M}_{n \times n}$ is defined to be the sum of its diagonal entries: $\operatorname{tr} A=a_{11}+a_{22}+\cdots+a_{n n}$. (a) Compute the trace of (i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}1 & 3 & 2 \\ -1 & 0 & 1 \\ -4 & 3 & -1\end{array}\right)$
(b) Prove that $\operatorname{tr}(A+B)=\operatorname{tr} A+\operatorname{tr} B$. (c) Prove that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$. (d) Prove that the commutator matrix $C=A B-B A$ has zero trace: $\operatorname{tr} C=0$. (e) Is part (c) valid if $A$ has size $m \times n$ and $B$ has size $n \times m$ ? (f) Prove that $\operatorname{tr}(A B C)=\operatorname{tr}(C A B)=\operatorname{tr}(B C A)$. On the other hand, find an example where $\operatorname{tr}(A B C) \neq \operatorname{tr}(A C B)$.

Chris Trentman
Chris Trentman
Numerade Educator
00:56

Problem 31

Suppose $A$ is regular. (a) Show that the matrix obtained by multiplying each column of $A$ by the sign of its pivot is also regular and, moreover, has all positive pivots.
(b) Show that the matrix obtained by multiplying each row of $A$ by the sign of its pivot is also regular and has all positive pivots.
(c) Check these results in the particular case $A=\left(\begin{array}{rrr}-2 & 2 & 1 \\ 1 & 0 & 1 \\ 4 & 2 & 3\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator
05:52

Problem 31

Solve the following systems of linear equations by computing the inverses of their coefficient matrices.
(a)
$$
\begin{aligned}
& x+2 y=1, \\
& x-2 y=-2 .
\end{aligned}
$$
(b)
$$
\begin{aligned}
3 u-2 v & =2, \\
u+5 v & =12 .
\end{aligned}
$$
(c)
$$
x-y+3 z=3,
$$
$$
y+5 z=3,
$$
(d)
$$
x-2 y+z=2 .
$$
$$
\begin{gathered}
=-2, \quad(d) x-y+3 z=-1, \\
=2 . \quad-2 x+3 y=5 . \\
x-2 y+z+2 u=-2, \\
x-y+z-u=3, \\
2 x-y+z+u=3,
\end{gathered}
$$
$$
x+y=4,
$$
(e)
$$
\begin{aligned}
2 x+7 y-2 z & =5, \\
-x-5 y+2 z & =-7 .
\end{aligned}
$$
(f)
$$
\begin{aligned}
2 x+3 y-w & =11, \\
-y-z+w & =-7, \\
z-w & =6 .
\end{aligned}
$$
(g)
$$
-x+3 y-2 z-u=2 .
$$

Ziya Ogron
Ziya Ogron
Numerade Educator

Problem 32

Prove that matrix multiplication is associative: $A(B C)=(A B) C$ when defined.

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07:24

Problem 32

Given the $L U$ factorizations you calculated in Exercise 1.3.21, solve the associated linear systems $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector with all entries equal to 1 .
CunDoong ThanCong com
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1 Linear Algebraic Systems

Bobby Barnes
Bobby Barnes
University of North Texas
01:13

Problem 32

For each of the nonsingular matrices in Exercise 1.5.25, use your computed inverse to solve the associated linear system $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector of the appropriate size that has all 1's as its entries.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
03:03

Problem 33

Justify the following alternative formula for multiplying a matrix $A$ and a column vector $\mathbf{x}$ :
$$
A \mathbf{x}=x_1 \mathbf{c}_1+x_2 \mathbf{c}_2+\cdots+x_n \mathbf{c}_n,
$$
where $\mathbf{c}_1, \ldots, \mathbf{c}_n$ are the columns of $A$ and $x_1, \ldots, x_n$ the entries of $\mathbf{x}$.

Lucas Finney
Lucas Finney
Numerade Educator
01:07

Problem 33

In each of the following problems, find the $A=L U$ factorization of the coefficient matrix, and then use Forward and Back Substitution to solve the corresponding linear systems $A \mathbf{x}=\mathbf{b}_j$ for each of the indicated right-hand sides:
(a) $A=\left(\begin{array}{rr}-1 & 3 \\ 3 & 2\end{array}\right), \mathbf{b}_1=\left(\begin{array}{r}1 \\ -1\end{array}\right), \mathbf{b}_2=\left(\begin{array}{l}2 \\ 5\end{array}\right), \mathbf{b}_3=\left(\begin{array}{l}0 \\ 3\end{array}\right)$.
(b) $A=\left(\begin{array}{rrr}-1 & 1 & -1 \\ 1 & 1 & 1 \\ -1 & 1 & 2\end{array}\right), \mathbf{b}_1=\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right), \mathbf{b}_2=\left(\begin{array}{r}-3 \\ 0 \\ 2\end{array}\right)$.
(c) $A=\left(\begin{array}{rrr}9 & -2 & -1 \\ -6 & 1 & 1 \\ 2 & -1 & 0\end{array}\right), \mathbf{b}_1=\left(\begin{array}{r}2 \\ -1 \\ 0\end{array}\right), \mathbf{b}_2=\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right)$.
(d) $A=\left(\begin{array}{rrr}2.0 & .3 & .4 \\ .3 & 4.0 & .5 \\ .4 & .5 & 6.0\end{array}\right), \mathbf{b}_1=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), \mathbf{b}_2=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right), \mathbf{b}_3=\left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)$.
(e) $A=\left(\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 2 & 3 & -1 \\ -1 & 3 & 2 & 2 \\ 0 & -1 & 2 & 1\end{array}\right), \mathbf{b}_1=\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 1\end{array}\right), \mathbf{b}_2=\left(\begin{array}{r}0 \\ -1 \\ 0 \\ 1\end{array}\right)$.
(f) $A=\left(\begin{array}{rrrr}1 & -2 & 0 & 2 \\ 4 & 1 & -1 & -1 \\ -8 & -1 & 2 & 1 \\ -4 & -1 & 1 & 2\end{array}\right), \mathbf{b}_1=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right), \mathbf{b}_2=\left(\begin{array}{r}3 \\ 0 \\ -1 \\ 2\end{array}\right), \quad \mathbf{b}_3=\left(\begin{array}{r}2 \\ 3 \\ -2 \\ 1\end{array}\right)$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 33

Produce the $L D V$ or a permuted $L D V$ factorization of the following matrices:
(a) $\left(\begin{array}{rr}1 & 2 \\ -3 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}0 & 4 \\ -7 & 2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}2 & 1 & 2 \\ 2 & 4 & -1 \\ 0 & -2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 1 & 5 \\ 1 & 1 & -2 \\ 2 & -1 & 3\end{array}\right)$,
(e) $\left(\begin{array}{lll}2 & -3 & 2 \\ 1 & -1 & 1 \\ 1 & -1 & 2\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}1 & -1 & 1 & 2 \\ 1 & -4 & 1 & 5 \\ 1 & 2 & -1 & -1 \\ 3 & 1 & 1 & 6\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}1 & 0 & 2 & -3 \\ 2 & -2 & 0 & 1 \\ 1 & -2 & -2 & -1 \\ 0 & 1 & 1 & 2\end{array}\right)$.

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02:29

Problem 34

The basic definition of matrix multiplication $A B$ tells us to multiply rows of $A$ by columns of $B$. Remarkably, if you suitably interpret the operation, you can also compute $A B$ by multiplying columns of $A$ by rows of $B$ ! Suppose $A$ is an $m \times n$ matrix with columns $\mathbf{c}_1, \ldots, \mathbf{c}_n$. Suppose $B$ is an $n \times p$ matrix with rows $\mathbf{r}_1, \ldots, \mathbf{r}_n$. Then we claim that
$$
A B=\mathbf{c}_1 \mathbf{r}_1+\mathbf{c}_2 \mathbf{r}_2+\cdots+\mathbf{c}_n \mathbf{r}_n \text {, }
$$
where each summand is a matrix of size $m \times p$. (a) Verify that the particular case
$$
\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right)\left(\begin{array}{rr}
0 & -1 \\
2 & 3
\end{array}\right)=\left(\begin{array}{l}
1 \\
3
\end{array}\right)\left(\begin{array}{ll}
0 & -1
\end{array}\right)+\left(\begin{array}{l}
2 \\
4
\end{array}\right)\left(\begin{array}{ll}
2 & 3
\end{array}\right)=\left(\begin{array}{rr}
0 & -1 \\
0 & -3
\end{array}\right)+\left(\begin{array}{rr}
4 & 6 \\
8 & 12
\end{array}\right)=\left(\begin{array}{ll}
4 & 5 \\
8 & 9
\end{array}\right)
$$
agrees with the usual method for computing the matrix product. (b) Use this method to compute the matrix products $(i)\left(\begin{array}{rr}-2 & 1 \\ 3 & 2\end{array}\right)\left(\begin{array}{rr}1 & -2 \\ 1 & 0\end{array}\right)$,
(ii)
$$
\left(\begin{array}{rrr}
1 & -2 & 0 \\
-3 & -1 & 2
\end{array}\right)\left(\begin{array}{rr}
2 & 5 \\
-3 & 0 \\
1 & -1
\end{array}\right)
$$
(iii) $\left(\begin{array}{rrr}3 & -1 & 1 \\ -1 & 2 & 1 \\ 1 & 1 & -5\end{array}\right)\left(\begin{array}{rrr}2 & 3 & 0 \\ 3 & -1 & 4 \\ 0 & 4 & 1\end{array}\right)$, and verify that you get the same answer as that obtained by the traditional method. (c) Explain why (1.13) is a special case of (1.14). (d) Prove that (1.14) gives the correct formula for the matrix product.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:05

Problem 34

Using the $L D V$ factorization for the matrices you found in parts (a-g) of Exercise 1.5.33, solve the corresponding linear systems $A \mathbf{x}=\mathbf{b}$, for the indicated vector $\mathbf{b}$.
(a) $\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{l}-1 \\ -2\end{array}\right)$,
(c) $\left(\begin{array}{r}1 \\ -3 \\ 2\end{array}\right)$,
(d) $\left(\begin{array}{r}-1 \\ 4 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{r}-1 \\ -2 \\ 5\end{array}\right)$,
(f) $\left(\begin{array}{r}2 \\ -9 \\ 3 \\ 4\end{array}\right)$,
(g) $\left(\begin{array}{r}6 \\ -4 \\ 0 \\ -3\end{array}\right)$

Heather Zimmers
Heather Zimmers
Numerade Educator
05:31

Problem 35

Matrix polynomials. Let $p(x)=c_n x^n+c_{n-1} x^{n-1}+\cdots+c_1 x+c_0$ be a polynomial function. If $A$ is a square matrix, we define the corresponding matrix polynomial $p(A)=$ $c_n A^n+c_{n-1} A^{n-1}+\cdots+c_1 A+c_0 I$; the constant term becomes a scalar multiple of the identity matrix. For instance, if $p(x)=x^2-2 x+3$, then $p(A)=A^2-2 A+3 \mathrm{I}$. (a) Write out the matrix polynomials $p(A), q(A)$ when $p(x)=x^3-3 x+2, q(x)=2 x^2+1$. (b) Evaluate $p(A)$ and $q(A)$ when $A=\left(\begin{array}{rr}1 & 2 \\ -1 & -1\end{array}\right)$. (c) Show that the matrix product $p(A) q(A)$ is the matrix polynomial corresponding to the product polynomial $r(x)=p(x) q(x)$. (d) True or false: If $B=p(A)$ and $C=q(A)$, then $B C=C B$. Check your answer in the particular case of part (b).

Elham Kordzadeh
Elham Kordzadeh
Numerade Educator
00:52

Problem 36

A block matrix has the form $M=\left(\begin{array}{ll}A & B \\ C & D\end{array}\right)$ in which $A, B, C, D$ are matrices with respective sizes $i \times k, i \times l, j \times k, j \times l$. (a) What is the size of $M$ ?
(b) Write out the block matrix $M$ when $A=\left(\begin{array}{l}1 \\ 3\end{array}\right), B=\left(\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right), C=\left(\begin{array}{r}1 \\ -2 \\ 1\end{array}\right), D=\left(\begin{array}{rr}1 & 3 \\ 2 & 0 \\ 1 & -1\end{array}\right)$.
(c) Show that if $N=\left(\begin{array}{cc}P & Q \\ n\end{array}\right)$ is a block matrix whose blocks have the same size as those

Dharmendra Jain
Dharmendra Jain
Numerade Educator
05:44

Problem 37

The matrix $S$ is said to be a square root of the matrix $A$ if $S^2=A$. (a) Show that $S=\left(\begin{array}{rr}1 & 1 \\ 3 & -1\end{array}\right)$ is a square root of the matrix $A=\left(\begin{array}{ll}4 & 0 \\ 0 & 4\end{array}\right)$. Can you find another square root of $A$ ? (b) Explain why only square matrices can have a square root. (c) Find all real square roots of the $2 \times 2$ identity matrix $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$.
(d) Does $-\mathrm{I}=\left(\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}\right)$ have a real square root?

Nick Johnson
Nick Johnson
Numerade Educator