Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases} w - x + 2y \quad\quad = -3 \\ \quad\quad x - y + z = 4 \\ -w + x - y + 2z = 2 \\ \quad\quad -x + y - 2z = -4 \end{cases} \\ \text{The inverse of } \begin{bmatrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \end{bmatrix} \text{ is } \begin{bmatrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \end{bmatrix}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} x + y + z = 4 \\ x - 2y + z = 7 \\ x + 3y + 2z = 4 \end{cases}$

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A - C

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \end{cases}$

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

$\begin{cases} 2 \ln w + \ln x + 3 \ln y - 2 \ln z = -6 \\ 4 \ln w + 3 \ln x + \ln y - \ln z = -2 \\ \ln w + \ln x + \ln y + \ln z = -5 \\ \ln w + \ln x - \ln y - \ln z = 5 \end{cases}$

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A(BC)

In Exercises 43–44, (a) Write each linear system as a matrix equation in the form AX = B (b) Solve the system using the inverse that is given for the coefficient matrix.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

In Exercises 46–51, evaluate each determinant.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

Evaluate each determinant in Exercises 49–52. $\begin{vmatrix} 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \end{vmatrix}$

Evaluate each determinant.

Evaluate each determinant in Exercises 49–52. $\begin{vmatrix} -2 & -3 & 3 & 5 \\ 1 & -4 & 0 & 0 \\ 1 & 2 & 2 & -3 \\ 2 & 0 & 1 & 1 \end{vmatrix}$

Evaluate each determinant.

In Exercises 53–54, evaluate each determinant. $\begin{vmatrix} \begin{vmatrix} 3 & 1 \\ -2 & 3 \end{vmatrix} & \begin{vmatrix} 7 & 0 \\ 1 & 5 \end{vmatrix} \\ \begin{vmatrix} 3 & 0 \\ 0 & 7 \end{vmatrix} & \begin{vmatrix} 9 & -6 \\ 3 & 5 \end{vmatrix} \end{vmatrix}$

Use Cramer's Rule to solve each system.

$\{\begin{array}{c}7x+2y=0\\ 2x+y=-3\end{array}$

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

In Exercises 57–60, solve each equation for x.

In Exercises 57–60, solve each equation for x.

Solve the system