Chapter 12, Supplementary notes on matrices and linear systems Video Solutions, Linear Algebra

Problem 1

In each of the following, find matrices $A, x$, and $b$ such that the given system of linear equations can be expressed as the single matrix equation $A x=b$.
$\left.\begin{array}{r}2 x_{1}-3 x_{2}+5 x_{3}=7 \\ \text { (a) } 9 x_{1}-x_{2}+x_{3}=-1 \\ x_{1}+5 x_{2}+4 x_{3}=0\end{array}\right\}$

Vysakh M

Numerade Educator

03:36

Problem 2

In each of the following, express the matrix equation as a system of linear equations.
(a) $\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]=\left[\begin{array}{c}2 \\ -1 \\ 4\end{array}\right]$
$$
\text { (b) }\left[\begin{array}{cccc}
3 & -2 & 0 & 1 \\
5 & 0 & 2 & -2 \\
3 & 1 & 4 & 7 \\
-2 & 5 & 1 & 6
\end{array}\right]\left[\begin{array}{l}
w \\
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

John Piaszynski

Numerade Educator

11:03

Problem 3

Suppose that $A, B, C, D$, and $E$ are matrices over $\mathbb{F}$ having the following sizes:
$A$ is $4 \times 5, B$ is $4 \times 5, C$ is $5 \times 2, D$ is $4 \times 2$,
Determine whether the following matrix expressions are defined, and, for those that are defined, determine the size of the resulting matrix.
(a) $B A$
(b) $A C+D$
(c) $A E+B$
(d) $A B+B$
(e) $E(A+B)$
$(f) E(A C)$

Willis James

Numerade Educator

11:03

Problem 4

Suppose that $A, B, C, D$, and $E$ are the following matrices:
$$
\begin{array}{c}
A=\left[\begin{array}{cc}
3 & 0 \\
-1 & 2 \\
1 & 1
\end{array}\right], B=\left[\begin{array}{cc}
4 & -1 \\
0 & 2
\end{array}\right], C=\left[\begin{array}{ccc}
1 & 4 & 2 \\
3 & 1 & 5
\end{array}\right] \\
D=\left[\begin{array}{ccc}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right], \text { and } E=\left[\begin{array}{ccc}
6 & 1 & 3 \\
-1 & 1 & 2 \\
4 & 1 & 3
\end{array}\right]
\end{array}
$$
Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix.
(a) $D+E$
(b) $D-E(c) 5 A(d)-7 C$
(e) $2 B-C$
$(f) 2 E-2 D(g)-3(D+2 E)$
$(h) A-A$
(i) $A B$
(j) $B A$
$(k)(3 E) D$
$(l)(A B) C(m) A(B C)(n)(4 B) C+2 B$
(o) $D-3 E$
$(p) C A+2 E$
$(q) 4 E-D(r) D D$

Willis James

Numerade Educator

09:25

Problem 5

Suppose that $A, B$, and $C$ are the following matrices and that $a=4$ and $b=7$.
$$
A=\left[\begin{array}{ccc}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right], B=\left[\begin{array}{ccc}
6 & 1 & 3 \\
-1 & 1 & 2 \\
4 & 1 & 3
\end{array}\right], \text { and } C=\left[\begin{array}{ccc}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right]
$$
Verify computationally that
(a) $A+(B+C)=(A+B)+C \quad(b)(A B) C=A(B C)$
(c) $(a+b) C=a C+b C$
(d) $a(B-C)=a B-a C$
$(e) a(B C)=(a B) C=B(a C) \quad(f) A(B-C)=A B-A C$
$(g)(B+C) A=B A+C A \quad(h) a(b C)=(a b) C$
$($ i) $B-C=-C+B$

Carole Wastog

Numerade Educator

16:42

Problem 6

Suppose that $A$ is the matrix
$$
A=\left[\begin{array}{ll}
3 & 1 \\
2 & 1
\end{array}\right]
$$
Compute $p(A)$, where $p(z)$ is given by
$$
\begin{array}{l}
(a) p(z)=z-2 \quad(b) p(z)=2 z^{2}-z+1 \\
(c) p(z)=z^{3}-2 z+4 \quad(d) p(z)=z^{2}-4 z+1
\end{array}
$$

Kyra Mcdermott

Numerade Educator

00:57

Problem 7

Define matrices $A, B, C, D$, and $E$ by
$$
\begin{array}{c}
A=\left[\begin{array}{ll}
3 & 1 \\
2 & 1
\end{array}\right], B=\left[\begin{array}{cc}
4 & -1 \\
0 & 2
\end{array}\right], C=\left[\begin{array}{ccc}
2 & -3 & 5 \\
9 & -1 & 1 \\
1 & 5 & 4
\end{array}\right], \\
D=\left[\begin{array}{ccc}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right], \text { and } E=\left[\begin{array}{ccc}
6 & 1 & 3 \\
-1 & 1 & 2 \\
4 & 1 & 3
\end{array}\right]
\end{array}
$$
(a) Factor each matrix into a product of elementary matrices and an RREF matrix.
(b) Find, if possible, the LU-factorization of each matrix.
(c) Determine whether or not each of these matrices is invertible, and, if possible, compute the inverse.

Mark Augustyn

Numerade Educator

07:55

Problem 8

Suppose that $A, B, C, D$, and $E$ are the following matrices:
$$
\begin{aligned}
A=\left[\begin{array}{cc}
3 & 0 \\
-1 & 2 \\
1 & 1
\end{array}\right], B=\left[\begin{array}{cc}
4 & -1 \\
0 & 2
\end{array}\right], C=\left[\begin{array}{ccc}
1 & 4 & 2 \\
3 & 1 & 5
\end{array}\right], \\
D=\left[\begin{array}{cll}
1 & 5 & 2 \\
-1 & 0 & 1 \\
3 & 2 & 4
\end{array}\right], \text { and } E=\left[\begin{array}{ccc}
6 & 1 & 3 \\
-1 & 1 & 2 \\
4 & 1 & 3
\end{array}\right]
\end{aligned}
$$
Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix.
(a) $2 A^{T}+C$
(b) $D^{T}-E^{T}$
(c) $(D-E)^{T}$
(d) $B^{T}+5 C^{T}$
$(e) \frac{1}{2} C^{T}-\frac{1}{4} A(f) B B^{T}$
(g) $3 E^{T}-3 D^{T}$
$(h)\left(2 E^{T}-3 D^{T}\right)^{T}$
(i) $C C^{T}$
$\begin{array}{lll}(j)(D A)^{T} & (k)\left(C^{T} B\right) A^{T} & (l)\left(2 D^{T}-E\right) A\end{array}$
$(m)\left(B A^{T}-2 C\right)^{T} \quad(n)$
(o) $D^{T} E^{T}-(E D)^{T}$
$(p) \operatorname{trace}\left(D D^{T}\right)$
$(q) \operatorname{trace}\left(4 E^{T}-D\right)(r) \operatorname{trace}\left(C^{T} A^{T}+2 E^{T}\right)$

Brandon Collins

Numerade Educator