Chapter 7

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 1 & 0 \\ 3 & -4 \end{array}\right| $$

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{cc} -2 & \frac{1}{2} \\ 3 & -1 \end{array}\right], \quad B=\left[\begin{array}{ll} -2 & -1 \\ -6 & -4 \end{array}\right] $$

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{l} x+5 y+8 z=19 \\ 12 x+3 y=4 \\ 3 x+4 y+9 z=-7 \end{array} $$

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{cc}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{cc}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{cc}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{cc}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{cc}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) $$ B-E $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{10 x+47}{x^{2}+7 x+10} $$

For the following exercises, solve the system of nonlinear equations using substitution. $$ \begin{array}{r} y=-x \\ x^{2}+y^{2}=9 \end{array} $$

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} x-y &=0 \\ x-z &=5 \\ x-y+z=-1 \end{aligned} \quad \text { and }(4,4,-1) $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 10 & 20 \\ 0 & -10 \end{array}\right| $$

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 1 & 1 \end{array}\right], \quad B=\frac{1}{2}\left[\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{array}\right] $$

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{r} 6 x+12 y+16 z=4 \\ 19 x-5 y+3 z=-9 \\ x+2 y=-8 \end{array} $$

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{cc}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{cc}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{cc}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{cc}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{cc}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) $$ C+F $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{x}{6 x^{2}+25 x+25} $$

For the following exercises, solve the system of nonlinear equations using substitution. $$ \begin{array}{r} x=2 \\ x^{2}-y^{2}=9 \end{array} $$

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} -x-y+2 z &=3 \\ 5 x+8 y-3 z &=4 \quad \text { and }(4,1,-7) \\ -x+3 y-5 z &=-5 \end{aligned} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{cc} 10 & 0.2 \\ 5 & 0.1 \end{array}\right| $$

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 0 & 2 \\ 1 & 6 & 9 \end{array}\right], \quad B=\frac{1}{4}\left[\begin{array}{ccc} 6 & 0 & -2 \\ 17 & -3 & -5 \\ -12 & 2 & 4 \end{array}\right] $$

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rr|r} -2 & 5 & 5 \\ 6 & -18 & 26 \end{array}\right] $$

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{cc}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{cc}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{cc}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{cc}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{cc}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) $$ D-B $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{32 x-11}{20 x^{2}-13 x+2} $$

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} 4 x^{2}-9 y^{2}=36 \\ 4 x^{2}+9 y^{2}=36 \end{array} $$

For the following exercises, solve each system by elimination. $$ \begin{array}{r} 3 x-4 y+2 z=-15 \\ 2 x+4 y+z=16 \\ 2 x+3 y+5 z=20 \end{array} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 6 & -3 \\ 8 & 4 \end{array}\right| $$

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{ccc} 3 & 8 & 2 \\ 1 & 1 & 1 \\ 5 & 6 & 12 \end{array}\right], \quad B=\frac{1}{36}\left[\begin{array}{ccc} -6 & 84 & -6 \\ 7 & -26 & 1 \\ -1 & -22 & 5 \end{array}\right] $$

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rr|r} 3 & 4 & 10 \\ 10 & 17 & 439 \end{array}\right] $$

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ 5 A $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{x+1}{x^{2}+7 x+10} $$

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} x^{2}+y^{2}=25 \\ x^{2}-y^{2}=1 \end{array} $$

For the following exercises, solve each system by elimination. $$ \begin{array}{c} 5 x-2 y+3 z=20 \\ 2 x-4 y-3 z=-9 \\ x+6 y-8 z=21 \end{array} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} -2 & -3 \\ 3.1 & 4,000 \end{array}\right| $$

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{cc} 3 & -2 \\ 1 & 9 \end{array}\right] $$

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rrr|r} 3 & 2 & 0 & 3 \\ -1 & -9 & 4 & -1 \\ 8 & 5 & 7 & 8 \end{array}\right] $$

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ 3 B $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{5 x}{x^{2}-9} $$

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} 2 x^{2}+4 y^{2}=4 \\ 2 x^{2}-4 y^{2}=25 x-10 \end{array} $$

For the following exercises, solve each system by elimination. $$ \begin{aligned} 5 x+2 y+4 z &=9 \\ -3 x+2 y+z &=10 \\ 4 x-3 y+5 z &=-3 \end{aligned} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} -1.1 & 0.6 \\ 7.2 & -0.5 \end{array}\right| $$

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

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rrr|r} 8 & 29 & 1 & 43 \\ -1 & 7 & 5 & 38 \\ 0 & 0 & 3 & 10 \end{array}\right] $$

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ -2 B $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{10 x}{x^{2}-25} $$

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} y^{2}-x^{2}=9 \\ 3 x^{2}+2 y^{2}=8 \end{array} $$

For the following exercises, solve each system by elimination. $$ \begin{array}{l} 4 x-3 y+5 z=31 \\ -x+2 y+4 z=20 \\ x+5 y-2 z=-29 \end{array} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3 \end{array}\right| $$

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{cc} -3 & 7 \\ 9 & 2 \end{array}\right] $$

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rrr|r} 4 & 5 & -2 & 12 \\ 0 & 1 & 58 & 2 \\ 8 & 7 & -3 & -5 \end{array}\right] $$

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ -4 C $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{6 x}{x^{2}-4} $$

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} x^{2}+y^{2}+\frac{1}{16}=2500 \\ y=2 x^{2} \end{array} $$

For the following exercises, solve each system by elimination. $$ \begin{aligned} 5 x-2 y+3 z &=4 \\ -4 x+6 y-7 z &=-1 \\ 3 x+2 y-z &=4 \end{aligned} $$

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} -1 & 4 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & -3 \end{array}\right| $$