Find the indicated matrices using a calculator.

$$A=\begin{bmatrix} 6 & -3 \\ 4 & -5 \end{bmatrix}$$

$$B=\begin{bmatrix} 3 & 12 \\ -9 & -6 \end{bmatrix}$$

$$C=\begin{bmatrix} -1 & 4 & -7 \\ 2 & -6 & 11 \end{bmatrix}$$

$$D=\begin{bmatrix} 7 & 9 & -6 \\ -4 & 0 & 8 \end{bmatrix}$$

$$A - B - 2C$$

Solve the given systems of equations by Gaussian elimination. Use those that are indicated from Exercises 31-38. Exercise 32

Solve the given systems of equations by determinants. Evaluate by expansion by minors.

$$\begin{array}{l} 2p+3r+s=4 \\ p-2r-3s+4t=-1 \\ 3p+r+s-5t=3 \\ -p+2r+s+3t=2 \end{array}$$

Find $CA^{-1}$

$$C=\left[ \begin{array}{rrr} 5 & -1 & 0 \\[6pt] 2 & -2 & 1 \\[6pt] -3 & 0 & 4 \end{array} \right]$$

$$A\left[ \begin{array}{rrr} 1 & 0 & 2 \\[6pt] -1 & 2 & 3 \\[6pt] 1 & -1 & 0 \end{array} \right]$$

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A. $A^2=A\times A$.

$$A=\left[\begin{array}{rrr} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 &-4 \end{array} \right]\text{ } B=\left[\begin{array}{rrr} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{array} \right] \text{ } C=\left[\begin{array}{rrr} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 &-4 \end{array} \right]$$

Show that B^3=B.

Use matrices A, B, C, and D to find the indicated matrices. If the operations cannot be performed, explain why.

$$A=\begin{bmatrix} 6 & -3 \\ 4 & -5 \end{bmatrix} \qquad B=\begin{bmatrix} 3 & 12 \\ -9 & -6\end{bmatrix}$$

$$C=\begin{bmatrix} -1 & -4 & -7 \\ 2 & -6 & 11 \end{bmatrix} \qquad D=\begin{bmatrix} 7 & 9 & -6 \\ -4 & 0 & 8 \end{bmatrix}$$

$$-6B-4A$$

Solve the given systems of equations by determinants. Evaluate by expansion by minors.

$$\begin{array}{l} x+2y-z=6\\ y-2z-3t=-5 \\ 3x-2y=t=2 \\ 2x+y+z-t=0 \end{array}$$

Forces F_1 and F_2 hold up a beam that weighs 2540 N, as shown in Fig. 16.14. The equations used to find the forces are

$$\begin{array}{l}A\sin 47.2^{\circ}+B\sin 64.4^{\circ}=2540 \\ A\cos 47.2^{\circ}-B\cos 64.4^{\circ}=0 \end{array}$$

Find the magnitude of each force.

Set up systems of equations and solve by Gaussian elimination. Three machines together produce 650 parts each hour. Twice the production of the second machine is 10 parts/h more than the sum of the production of the other two machines. If the first operates for 3.00 h and the others operate for 2.00, 1550 parts are produced. Find the production rate of each machine.

Solve the given systems of equations by Gaussian elimination. Use those that are indicated from Exercises 31-38. Exercise 31

Find $CA^{-1}$

$$C=\left[ \begin{array}{rrr} 5 & -1 & 0 \\[6pt] 2 & -2 & 1 \\[6pt] -3 & 0 & 4 \end{array} \right]$$

$$A\left[ \begin{array}{rrr} 1 & -1 & 1 \\[6pt] 0 & -2 & 1 \\[6pt] -2 & -3 & 0 \end{array} \right]$$

Set up systems of equations and solve by Gaussian elimination. The voltage across an electric resistor equals the current (in A) times the resistance (in $\ohm$). If a current of 3.00 A passes through each of the two resistors, the sum of the voltages is 10.5 V. If 2.00 A passes through the first resistor and 4.00 A passes through the second resistor, the sum of the voltages is 13.0 V. Find the resistances.

Solve the given systems of equations using the inverse of the coefficient matrix.

$$\begin{array}{l}3x+2y+z=2 \\ 2x+3y-6z=3 \\ x+3y+3z=1 \end{array}$$

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix A. $A^2=A\times A$.

$$A=\left[\begin{array}{rrr} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 &-4 \end{array} \right]\text{ } B=\left[\begin{array}{rrr} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{array} \right] \text{ } C=\left[\begin{array}{rrr} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 &-4 \end{array} \right]$$

Show that C^2=O.