7. Systems of Equations & Matrices

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\)2w + x - y = 3 \(\w\) - 3x + 2y = -4 \\3w + x - 3y + z = 1 \(\w\) + 2x - 4y - z = -2\(\end{cases}\)$

Write the partial fraction decomposition of each rational expression. (4x²+3x+14)/(x³ - 8)

Solve by eliminating variables:

Write the system of equations associated with each augmented matrix . Do not solve.

Perform each matrix row operation and write the new matrix.

$\(\begin{bmatrix}\)2 & -6 & 4 & \(\vert\) & 10 \\1 & 5 & -5 & \(\vert\) & 0 \\3 & 0 & 4 & \(\vert\) & 7\(\end{bmatrix}\]\quad\) \(\frac{1}{2}\)R_1$

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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

$\(\begin{cases}\)x + 2y = z - 1 \(\x\) = 4 + y - z \(\x\) + y - 3z = -2\(\end{cases}\)$

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x + 1)/(x + 2)^3

In Exercises 9 - 16, find the following matrices: a. A + B

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\)8x + 5y + 11z = 30 \\-x - 4y + 2z = 3 \\2x - y + 5z = 12\(\end{cases}\)$

Find the dimension of each matrix. Identify any square, column, or row matrices.

$\left[\begin{array}{c}2\\ 4\end{array}\right]$

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

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}\)$

a. Give the order of each matrix.

b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

$\(\begin{bmatrix}\)1 & -5 & \(\pi\) & e \\0 & 7 & -6 & -\(\pi\) \\-2 & \(\frac{1}{2}\) & 11 & -\(\frac{1}{5}\]\end{bmatrix}\)$

Find the dimension of each matrix. Identify any square, column, or row matrices.