Chapter 12

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -2 & -5 \\ -3 & -6 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} -3 & 4 \\ 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & 5 \\ 6 & -1 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} x+y>1 \\ x+y>3 \end{array}\right) $$

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -2 & 5 \\ -3 & 6 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} 1 & -3 \\ -4 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 7 & -3 \\ 4 & 5 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} y \geq x \\ x>-1 \end{array}\right) $$

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{r} -2 \\ 3 \\ -5 \end{array}\right], \quad B=\left[\begin{array}{ccc} 3 & -4 & -5 \end{array}\right] $$

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} -3 & 4 \\ 1 & -2 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} 5 & 0 \\ -2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} -3 & 6 \\ 4 & 1 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} y

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{r} 2 \\ -7 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -2 \\ 1 & 0 \\ -1 & 4 \end{array}\right] $$

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 2 & -4 \\ 1 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -2 \\ -3 & 6 \end{array}\right] $$

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{rrrr} 3 & -2 & 2 & -4 \\ 1 & 0 & -1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -3 & 1 & 4 \\ 5 & 2 & 0 \\ -4 & -1 & -2 \end{array}\right] $$

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 2 \\ -1 & -1 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} x \leq 3 \\ y \leq-1 \end{array}\right) $$

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{r} 3 \\ -4 \\ 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & -4 \end{array}\right] $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{ll} 4 & 3 \\ 2 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 6 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} -3 & -2 \\ -4 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ 4 & 5 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} y>-2 \\ x>1 \end{array}\right) $$

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{ll} 3 & -7 \end{array}\right], \quad B=\left[\begin{array}{r} 8 \\ -9 \end{array}\right] $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} 5 & -2 \\ 3 & 1 \end{array}\right], \quad B=\left[\begin{array}{l} 5 \\ 8 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} -2 & 3 \\ -1 & 7 \end{array}\right], \quad B=\left[\begin{array}{ll} -1 & -3 \\ -5 & -7 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} x+2 y>4 \\ x+2 y<2 \end{array}\right) $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} -3 & -4 \\ 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{r} 4 \\ -3 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & 1 \\ 5 & 2 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{rl} x & \geq 0 \\ y & \geq 0 \\ x+y & \leq 4 \\ 2 x+y & \leq 6 \end{array}\right) $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} 1 & 2 \\ 2 & -3 \end{array}\right] $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} 5 & 2 \\ -1 & -3 \end{array}\right], \quad B=\left[\begin{array}{r} 3 \\ -5 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} -8 & -5 \\ 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & -5 \\ 3 & 8 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{rl} x & \geq 0 \\ y & \geq 0 \\ x-y & \leq 5 \\ 4 x+7 y & \leq 28 \end{array}\right) $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} -4 & 2 \\ 7 & -5 \end{array}\right], \quad \bar{B}=\left[\begin{array}{l} -1 \\ -4 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} \frac{1}{2} & -\frac{1}{3} \\ \frac{1}{3} & \frac{1}{4} \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & -6 \\ 6 & -4 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{r} x \geq 0 \\ y \geq 0 \\ 2 x+y \leq 4 \\ 2 x-3 y \leq 6 \end{array}\right) $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} 0 & -3 \\ 2 & 9 \end{array}\right], \quad B=\left[\begin{array}{l} -\overline{3} \\ -6 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} \frac{1}{3} & -\frac{1}{2} \\ \frac{3}{2} & -\frac{2}{3} \end{array}\right], \quad B=\left[\begin{array}{cc} -6 & -18 \\ 12 & -12 \end{array}\right] $$

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{rl} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \geq 15 \\ 5 x+3 y & \geq 15 \end{array}\right) $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{ll} -2 & -3 \\ -5 & -6 \end{array}\right], \quad B=\left[\begin{array}{r} 5 \\ -2 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 5 & 6 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -2 \\ -\frac{2}{3} & \frac{5}{3} \end{array}\right] $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} -3 & 1 \\ 3 & -2 \end{array}\right] $$

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} -3 & -5 \\ 4 & -7 \end{array}\right], \quad B=\left[\begin{array}{r} -3 \\ -10 \end{array}\right] $$

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} -3 & -5 \\ 2 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & -\frac{5}{2} \\ 1 & \frac{3}{2} \end{array}\right] $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{array}\right] $$

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A B \text { and } B A \text {. } $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{c} 2 x+3 y=13 \\ x+2 y=8 \end{array}\right) $$

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A C \text { and } C A \text {. } $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 1 & -2 & 1 \\ -2 & 5 & 3 \\ 3 & -5 & 7 \end{array}\right] $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{rl} 4 x-3 y & =-23 \\ -3 x+2 y & =16 \end{array}\right) $$

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A D \text { and } D A \text {. } $$