Chapter 7

For the following exercises, use the matrices below to perform matrix multiplication. \(A=\left[\begin{array}{rr}-1 & 5 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rrr}3 & 6 & 4 \\ -8 & 0 & 12\end{array}\right], C=\left[\begin{array}{rr}4 & 10 \\ -2 & 6 \\ 5 & 9\end{array}\right], D=\left[\begin{array}{rrr}2 & -3 & 12 \\ 9 & 3 & 1 \\ 0 & 8 & -10\end{array}\right]\) \(C B\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{-24 x-27}{(4 x+5)^{2}} $$

For the following exercises, use any method to solve the system of nonlinear equations. $$ \begin{array}{c} 2 x^{3}-x^{2}=y \\ x^{2}+y=0 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{c} x+y+z=14 \\ 2 y+3 z=-14 \\ -16 y-24 z=-112 \end{array} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{ccc|} -\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{5} & -\frac{1}{6} & \frac{1}{7} \\ 0 & 0 & \frac{1}{8} \end{array}\right| $$

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{ccc} 1 & -2 & 3 \\ -4 & 8 & -12 \\ 1 & 4 & 2 \end{array}\right] $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -4 x-3 y=-2 \\ 3 x-5 y=-13 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(A+B-C\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{-24 x-27}{(6 x-7)^{2}} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} x^{2}+y^{2} &=9 \\ y &=3-x^{2} \end{aligned} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 5 x-3 y+4 z=-1 \\ -4 x+2 y-3 z=0 \\ -x+5 y+7 z=-11 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 2 x-3 y=-1 \\ 4 x+5 y=9 \end{array} $$

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \end{array}\right] $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -5 x+8 y=3 \\ 10 x+6 y=5 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(4 A+5 D\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{5-x}{(x-7)^{2}} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ x=3 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} x+y+z=0 \\ 2 x-y+3 z=0 \\ x-z=0 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 5 x-4 y=2 \\ -4 x+7 y=6 \end{array} $$

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} 3 x+4 y=12 \\ -6 x-8 y=-24 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(2 C+B\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{5 x+14}{2 x^{2}+12 x+18} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{c} x^{2}-y^{2}=9 \\ y=3 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 3 x+2 y-5 z=6 \\ 5 x-4 y+3 z=-12 \\ 4 x+5 y-2 z=15 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 6 x-3 y=2 \\ -8 x+9 y=-1 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} 5 x-6 y=-61 \\ 4 x+3 y=-2 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -60 x+45 y=12 \\ 20 x-15 y=-4 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(3 D+4 E\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{5 x^{2}+20 x+8}{2 x(x+1)^{2}} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ x-y=0 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{c} x+y+z=0 \\ 2 x-y+3 z=0 \\ x-z=1 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 2 x+6 y=12 \\ 5 x-2 y=13 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} 8 x+4 y=-100 \\ 3 x-4 y=1 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} 11 x+10 y=43 \\ 15 x+20 y=65 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(C-0.5 D\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{4 x^{2}+55 x+25}{5 x(3 x+5)^{2}} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{l} -x^{2}+y=2 \\ -4 x+y=-1 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{c} 3 x-\frac{1}{2} y-z=-\frac{1}{2} \\ 4 x+z=3 \\ -x+\frac{3}{2} y=\frac{5}{2} \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x+3 y=23 \\ 2 x-y=-1 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} 3 x-2 y=6 \\ -x+5 y=-2 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} 2 x-y=2 \\ 3 x+2 y=17 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(100 D-10 E\)

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{54 x^{3}+127 x^{2}+80 x+16}{2 x^{2}(3 x+2)^{2}} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{c} -x^{2}+y=2 \\ 2 y=-x \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{r} 6 x-5 y+6 z=38 \\ \frac{1}{5} x-\frac{1}{2} y+\frac{3}{5} z=1 \\ -4 x-\frac{3}{2} y-z=-74 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 10 x-6 y=2 \\ -5 x+8 y=-1 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} 5 x-4 y=-5 \\ 4 x+y=2.3 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -1.06 x-2.25 y=5.51 \\ -5.03 x-1.08 y=5.40 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A B\)