7. Systems of Equations & Matrices

Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x⁵ + 3x⁴ - 3x³ - 2x² + x)/(2x² + 5x + 2)

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

$\{\begin{array}{c}3x_1+5x_2-8x_3+5x_4=-8\\ x_1+2x_2-3x_3+x_4=-7\\ 2x_1+3x_2-7x_3+3x_4=-11\\ 4x_1+8x_2-10x_3+7x_4=-10\end{array}$

Write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

a. Give the order of each matrix.

b. If $A=\left[a_{ij}\right]$, identify $a_{32}$ and $a_{23}$, or explain why identification is not possible.

$\(\begin{bmatrix}\)4 & -7 & 5 \\-6 & 8 & -1\(\end{bmatrix}\)$

Solve each system in Exercises 5–18.

$\(\begin{cases}\)x + y + 2z = 11 \(\x\) + y + 3z = 14 \(\x\) + 2y - z = 5\(\end{cases}\)$

Write the augmented matrix for each system of linear equations.

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

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-x⁴ - 8x² + 3x - 10)/((x + 2)(x² + 4)²)

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\)w - 2x - y - 3z = -9 \(\w\) + x - y = 0 \\3w + 4x + z = 6 \\2x - 2y + z = 3\(\end{cases}\)$

Solve each system in Exercises 5–18. $\(\begin{cases}\)2x - 4y + 3z = 17 \(\x\) + 2y - z = 0 \\4x - y - z = 6\(\end{cases}\)$

Let and . Solve each matrix equation for X. 4A + 3B = - 2X

Write the partial fraction decomposition of each rational expression. x²/(x − 1)² (x + 1)

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. $\(\begin{bmatrix}\)x & 2y \(\z\) & 9\(\end{bmatrix}\)=\(\begin{bmatrix}\)4 & 12 \\3 & 9\(\end{bmatrix}\)$

How many rows and how many columns does this matrix have? What is its dimension?

Write the partial fraction decomposition of each rational expression. (6x-11)/(x − 1)²