7. Systems of Equations & Matrices

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

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\)5x + 12y + z = 10 \\2x + 5y + 2z = -1 \\(\x\) + 2y - 3z = 5\(\end{cases}\)$

Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x²+1) + 2x/(x²+1)².

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

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\(\end{cases}\)$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\(\frac{5x^2 - 6x + 7}{(x - 1)(x^2 + 1)}\)$

Write the partial fraction decomposition of each rational expression. 4x² - 7x - 3/(x³ -x)

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.

3x + 2y = -9

2x - 5y = -6

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

Perform each matrix row operation and write the new matrix.

$\(\begin{bmatrix}\)1 & -1 & 1 & 1 & \(\vert\) & 3 \\0 & 1 & -2 & -1 & \(\vert\) & 0 \\2 & 0 & 3 & 4 & \(\vert\) & 11 \\5 & 1 & 2 & 4 & \(\vert\) & 6\(\end{bmatrix}\[\quad\]\begin{array}{l}\)-2R_1 + R_3 \\-5R_1 + R_4\(\end{array}\)$

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

Perform each long division and write the partial fraction decomposition of the remainder term. (x⁵+2)/(x²-1)

Perform each long division and write the partial fraction decomposition of the remainder term. (x⁴-x²+2)/(x³-x²)

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