Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin

Chapter 2

Matrices and Systems of Linear Equations - all with Video Answers

Educators

Section 1

Matrices: Definitions and Notation

Problem 1

If $A=\left[\begin{array}{rrrr}1 & -2 & 3 & 2 \\ 7 & -6 & 5 & -1 \\ 0 & 2 & -3 & 4\end{array}\right],$ determine
(a) $a_{31}, a_{24}, a_{14}, a_{32}, a_{21},$ and $a_{34}$
(b) all pairs $(i, j)$ such that $a_{i j}=2$

Nez Nikoo

Nez Nikoo

Numerade Educator

Problem 2

$$^{165}=\left[\begin{array}{c}-1^{-1}-\frac{1}{-3} \\ -\frac{5}{-1} \\ -1 \\ 6 \\ -\frac{4}{4}\end{array}\right]$$
(a) $b_{12}, b_{33}, b_{41}, b_{43}, b_{51},$ and $b_{52}$
(b) all pairs $(i, j)$ such that $b_{i j}=-1$

Nez Nikoo

Numerade Educator

Problem 3

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$a_{11}=1, a_{21}=-1, a_{12}=5, a_{22}=3$$

Nez Nikoo

Numerade Educator

Problem 4

$$\begin{aligned}
&a_{11}=2, a_{12}=1, a_{13}=-1, a_{21}=0, a_{22}=4\\
&a_{23}=-2
\end{aligned}$$

Nez Nikoo

Numerade Educator

Problem 5

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$a_{11}=-1, a_{41}=-5, a_{31}=1, a_{21}=1$$

Nez Nikoo

Numerade Educator

Problem 6

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$\begin{array}{l}
a_{11}=1, a_{31}=2, a_{42}=-1, a_{32}=7, a_{13}=-2 \\
a_{23}=0, a_{33}=4, a_{21}=3, a_{41}=-4, a_{12}=-3 \\
a_{22}=6, a_{43}=5
\end{array}$$

Melissa Munoz

Numerade Educator

Problem 7

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$\begin{aligned}
&a_{12}=-1, a_{13}=2, a_{23}=3, a_{j i}=-a_{i j}, 1 \leq i \leq 3\\
&1 \leq j \leq 3
\end{aligned}$$

Nez Nikoo

Numerade Educator

Problem 8

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$a_{i j}=i-j, 1 \leq i \leq 4,1 \leq j \leq 4$$

Nez Nikoo

Numerade Educator

Problem 9

Write the matrix with the given elements In each case, specify the dimensions of the matrix.
$$a_{i j}=i+j, 1 \leq i \leq 4,1 \leq j \leq 4$$

Nez Nikoo

Numerade Educator

Problem 10

Determine $\operatorname{tr}(A)$ for the given matrix.
$$A=\left[\begin{array}{ll}
1 & 0 \\
2 & 3
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 11

Determine $\operatorname{tr}(A)$ for the given matrix.
$$A=\left[\begin{array}{lll}
1 & 2 & -1 \\
3 & 2 & -2 \\
7 & 5 & -3
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 12

Determine $\operatorname{tr}(A)$ for the given matrix.
$$A=\left[\begin{array}{lll}
2 & 0 & 1 \\
3 & 2 & 5 \\
0 & 1 & -5
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 13

Write the column vectors and row vectors of the given matrix.
$$A=\left[\begin{array}{rr}
1 & -1 \\
3 & 5
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 14

Write the column vectors and row vectors of the given matrix.
$$A=\left[\begin{array}{rrr}
1 & 3 & -4 \\
-1 & -2 & 5 \\
2 & 6 & 7
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 15

Write the column vectors and row vectors of the given matrix.
$$A=\left[\begin{array}{ccc}
2 & 10 & 6 \\
5 & -1 & 3
\end{array}\right]$$

Nez Nikoo

Numerade Educator

Problem 16

If $\mathbf{a}_{1}=\left[\begin{array}{lll}1 & 2\end{array}\right], \mathbf{a}_{2}=\left[\begin{array}{ll}3 & 4\end{array}\right],$ and $\mathbf{a}_{3}=\left[\begin{array}{ll}5 & 1\end{array}\right],$ write
the matrix
$$A=\left[\begin{array}{l}
\mathbf{a}_{1} \\
\mathbf{a}_{2} \\
\mathbf{a}_{3}
\end{array}\right]$$
and determine the column vectors of $A$

Nez Nikoo

Numerade Educator

Problem 17

If $\mathbf{a}_{1}=\left[\begin{array}{lllll}-2 & 0 & 4 & -1 & -1\end{array}\right]$ and
$\mathbf{a}_{2}=\left[\begin{array}{lllll}9 & -4 & -4 & 0 & 8\end{array}\right],$ write the matrix
$$A=\left[\begin{array}{l}
\mathbf{a}_{1} \\
\mathbf{a}_{2}
\end{array}\right]$$
and determine the column vectors of $A .$

Nez Nikoo

Numerade Educator

Problem 18

If
$$\mathbf{b}_{1}=\left[\begin{array}{r}-2 \\-6 \\3 \\-1 \\-2\end{array}\right] \quad \text { and } \quad \mathbf{b}_{2}=\left[\begin{array}{r}-4 \\-6 \\0 \\0 \\1\end{array}\right]$$
write the matrix $B=\left[\mathbf{b}_{1}, \mathbf{b}_{2}\right],$ and determine the row vectors of $B$.

Nez Nikoo

Numerade Educator

Problem 19

If
$$\begin{aligned}\mathbf{b}_{1}=&\left[\begin{array}{r}2 \\-1 \\4\end{array}\right], \quad \mathbf{b}_{2}=\left[\begin{array}{r}5 \\7 \\-6\end{array}\right] \\\mathbf{b}_{3}=\left[\begin{array}{l}0 \\
0 \\0\end{array}\right], & \mathbf{b}_{4}=\left[\begin{array}{l}1 \\2 \\3\end{array}\right]\end{aligned}$$
write the matrix $B=\left[\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}, \mathbf{b}_{4}\right],$ and determine
the row vectors of $B$

Nez Nikoo

Numerade Educator

Problem 20

If $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{p}$ are each column $q$ -vectors, what are the dimensions of the matrix that has $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{p}$ as its column vectors?

Diogo Caetano

Numerade Educator

Problem 21

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$3 \times 3$ diagonal matrix.

Nez Nikoo

Numerade Educator

Problem 22

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$4 \times 4$ upper triangular matrix.

Nez Nikoo

Numerade Educator

Problem 23

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$4 \times 4$ skew-symmetric matrix.

Nez Nikoo

Numerade Educator

Problem 24

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$3 \times 3$ upper triangular symmetric matrix.

Nez Nikoo

Numerade Educator

Problem 25

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$3 \times 3$ lower triangular skew-symmetric matrix.

Nez Nikoo

Numerade Educator

Problem 26

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.)
$3 \times 3$ symmetric and skew-symmetric matrix.

Nez Nikoo

Numerade Educator

Problem 27

Given an example of a matrix function of the specified form. (Many examples may be possible.)
$4 \times 2$ matrix function $A$ such that
$$A(0)=A(1) \neq A(2)$$

Nez Nikoo

Numerade Educator

Problem 28

Given an example of a matrix function of the specified form. (Many examples may be possible.)
$2 \times 3$ matrix function defined only for values of $t$ with $-2 \leq t<3$

Nez Nikoo

Numerade Educator

Problem 29

Given an example of a matrix function of the specified form. (Many examples may be possible.)
$2 \times 1$ matrix function $A$ that is nonconstant such that all elements of $A(t)$ are in [0,1] for every $t$ in $\mathbb{R}$

Nez Nikoo

Numerade Educator

Problem 30

Given an example of a matrix function of the specified form. (Many examples may be possible.)
$1 \times 5$ matrix function $A$ that is nonconstant such that all elements of $A(t)$ are positive for all $t$ in $\mathbb{R}$.

Nez Nikoo

Numerade Educator

Problem 31

Construct distinct matrix functions $A$ and $B$ defined on all of $\mathbb{R}$ such that $A(0)=B(0)$ and $A(1)=B(1)$

Nez Nikoo

Numerade Educator

Problem 32

Show that an $n \times n$ symmetric upper triangular matrix is diagonal. [Hint: This amounts to showing that if $\left.i \neq j, \text { then } a_{i j}=0 .\right]$

Victor Salazar

Numerade Educator

Problem 33

Show that if $A$ is an $n \times n$ matrix that is both symmetric and skew-symmetric, then every element of $A$ is zero. (Such a matrix is called a zero matrix.)

Nez Nikoo

Numerade Educator