7. Systems of Equations & Matrices

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

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

Use the given row transformation to change each matrix as indicated.

Determine if the given ordered triple is a solution of the system. $(2,-1,3)$

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

Perform each matrix row operation and write the new matrix.

$\(\begin{bmatrix}\)1 & 2 & 2 & \(\vert\) & 2 \\0 & 1 & -1 & \(\vert\) & 2 \\0 & 5 & 4 & \(\vert\) & 1\(\end{bmatrix}\)-5R_2 + R_3$

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x^2 - x - 15)/(x(x + 1)(x - 1))

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

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{6x^2-14x-27}{(x+2)(x-3{)}^2}$

Write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

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

Perform the indicated Row Operation.

SWAP $R_1\(\leftrightarrow\) R_2$

Write the partial fraction decomposition of each rational expression. 4/(2x² -5x -3)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)

Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?

Find the following matrices: - 3A + 2B

$A = \(\begin{bmatrix}\)3 & 1 & 1 \\-1 & 2 & 5\(\end{bmatrix}\), B = \(\begin{bmatrix}\)2 & -3 & 6 \\-3 & 1 & -4\(\end{bmatrix}\)$

Solve each system in Exercises 5–18. $\(\begin{cases}\)4x - y + 2z = 11 \\(\x\) + 2y - z = -1 \\2x + 2y - 3z = -1\(\end{cases}\)$