7. Systems of Equations & Matrices

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

x + y - 5z = -18

3x - 3y + z = 6

x + 3y - 2z = -13

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}3a - b - 4c = 3 \\2a - b + 2c = -8 \\a + 2b - 3c = 9\end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}2x + 2y + 7z = -1 \\2x + y + 2z = 2 \\4x + 6y + z = 15\end{cases}$

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}w + x + y + z = 4 \\2w + x - 2y - z = 0 \\w - 2x - y - 2z = -2 \\3w + 2x + y + 3z = 4\end{cases}$

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

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.

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)

Solve by eliminating variables:

Solve each system in Exercises 25–26. $\begin{cases}\frac{x + 3}{2} - \frac{y - 1}{2} + \frac{z + 2}{4} = \frac{3}{2} \\\frac{x - 5}{2} + \frac{y + 1}{3} - \frac{z}{4} = - \frac{25}{6} \\\frac{x - 3}{4} - \frac{y + 1}{2} + \frac{z - 3}{2} = - \frac{5}{2}\end{cases}$

Solve each system in Exercises 25–26. $\begin{cases}\frac{x + 2}{6} - \frac{y + 4}{3} + \frac{z}{2} = 0 \\\frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = \frac{9}{2} \\\frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{19}{4}\end{cases}$

Exercises 57–59 will help you prepare for the material covered in the next section. Solve: $\begin{cases}A + B = 3 \\2A - 2B + C = 17 \\4A - 2C = 14\end{cases}$