Chapter 5

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{ll}1 & -3 \\ 1 & -2\end{array}\right]\) and \(\left[\begin{array}{ll}-2 & 3 \\ -1 & 1\end{array}\right]\)

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(2 \times 3\), and \(B\) is of size \(3 \times 5\).

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) What is the size of \(A ?\) Of \(B ?\) Of \(C\) ? Of \(D ?\)

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrr|r}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2\end{array}\right]\)

Write the augmented matrix corresponding to each system of equations. \(2 x-3 y=7\) \(3 x+y=4\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} x-3 y &=-1 \\ 4 x+3 y &=11 \end{aligned}\)

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{ll}4 & 5 \\ 2 & 3\end{array}\right]\) and \(\left[\begin{array}{rr}\frac{3}{2} & -\frac{5}{2} \\ -1 & 2\end{array}\right]\)

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(3 \times 4\), and \(B\) is of size \(4 \times 3\).

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Find \(a_{14}, a_{21}, a_{31}\), and \(a_{43} .\)

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrr|r}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 1\end{array}\right]\)

Write the augmented matrix corresponding to each system of equations. \(\begin{aligned} 3 x+7 y-8 z &=5 \\ x &+3 z=&-2 \\ 4 x-3 y &=7 \end{aligned}\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 2 x-4 y &=5 \\ 3 x+2 y &=6 \end{aligned}\)

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{lll}3 & 2 & 3 \\ 2 & 2 & 1 \\ 2 & 1 & 1\end{array}\right]\) and \(\left[\begin{array}{rrr}-\frac{1}{3} & -\frac{1}{3} & \frac{4}{3} \\ 0 & 1 & -1 \\ \frac{2}{3} & -\frac{1}{3} & -\frac{2}{3}\end{array}\right]\)

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(1 \times 7\), and \(B\) is of size \(7 \times 1\).

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Find \(b_{13}, b_{31}\), and \(b_{43}\).

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{ll|l}1 & 0 & 2 \\ 0 & 1 & 4 \\ 0 & 0 & 0\end{array}\right]\)

Write the augmented matrix corresponding to each system of equations. \(\begin{aligned}-y+2 z &=6 \\ 2 x+2 y-8 z &=7 \\ 3 y+4 z &=0 \end{aligned}\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} x+4 y &=7 \\ \frac{1}{2} x+2 y &=5 \end{aligned}\)

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{rrr}2 & 4 & -2 \\ -4 & -6 & 1 \\ 3 & 5 & -1\end{array}\right]\) and \(\left[\begin{array}{rrr}\frac{1}{2} & -3 & -4 \\\ -\frac{1}{2} & 2 & 3 \\ -1 & 1 & 2\end{array}\right]\)

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(4 \times 4\), and \(B\) is of size \(4 \times 4\).

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Identify the row matrix. What is its transpose?

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{lll|l}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Write the augmented matrix corresponding to each system of equations. \(\begin{aligned} 3 x_{1}+2 x_{2} &=0 \\ x_{1}-x_{2}+2 x_{3} &=4 \\ 2 x_{2}-3 x_{3} &=5 \end{aligned}\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 3 x-4 y &=7 \\ 9 x-12 y &=14 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]\)

Let \(A\) be a matrix of size \(m \times n\) and \(B\) be a matrix of size \(s \times t\). Find conditions on \(m, n, s\), and \(t\) such that both matrix products \(A B\) and \(B A\) are defined.

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Identify the column matrix. What is its transpose?

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrr|r}1 & 0 & 1 & 4 \\ 0 & 1 & 0 & -2\end{array}\right]\)

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{rr|r}3 & 2 & -4 \\ 1 & -1 & 5\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} x+2 y &=7 \\ 2 x-y &=4 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right]\)

Find condition(s) on the size of a matrix \(A\) such that \(A^{2}\) (that is, \(A A\) ) is defined.

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Identify the square matrix. What is its transpose?

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrrr|r}1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & 2\end{array}\right]\)

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{rrr|r}0 & 3 & 2 & 4 \\ 1 & -1 & -2 & -3 \\ 4 & 0 & 3 & 2\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. . \(\begin{aligned} \frac{3}{2} x-2 y &=4 \\ x+\frac{1}{3} y &=2 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rr}3 & -3 \\ -2 & 2\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{ll}1 & 2 \\ 3 & 0\end{array}\right]\left[\begin{array}{r}1 \\ -1\end{array}\right]\)

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) What is the size of \(A ?\) Of \(B\) ? Of \(C\) ? Of \(D\) ?

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{llll|l}1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1\end{array}\right]\)

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{rrr|r}1 & 3 & 2 & 4 \\ 2 & 0 & 0 & 5 \\ 3 & -3 & 2 & 6\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 2 x-5 y &=10 \\ 6 x-15 y &=30 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{ll}4 & 2 \\ 6 & 3\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{rr}-1 & 3 \\ 5 & 0\end{array}\right]\left[\begin{array}{l}7 \\ 2\end{array}\right]\)

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{lll|l}2 & 3 & 1 & 6 \\ 4 & 3 & 2 & 5 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 5 x-6 y &=8 \\ 10 x-12 y &=16 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}2 & -3 & -4 \\ 0 & 0 & -1 \\ 1 & -2 & 1\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{rrr}3 & 1 & 2 \\ -1 & 2 & 4\end{array}\right]\left[\begin{array}{r}4 \\ 1 \\ -2\end{array}\right]\)

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(A+B\).

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrrr|r}1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)