7. Systems of Equations & Matrices

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 - C

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}5x + 8y - 6z = 14 \\3x + 4y - 2z = 8 \\x + 2y - 2z = 3\end{cases}$

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

$\begin{cases}3w - 4x + y + z = 9 \\w + x - y - z = 0 \\2w + x + 4y - 2z = 3 \\-w + 2x + y - 3z = 3\end{cases}$

Solve each problem. See Examples 5 and 9. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \$351. How many of each denomination of bill are there?

Write the augmented matrix for each system of linear equations.

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

Write the equations in standard form, then represent the system using an augmented matrix.

$3x+5y-9=0$

$8x=-4y+3$

Perform the indicated Row Operation.

SWAP $R_1\leftrightarrow R_2$

Perform the indicated Row Operation.

ADD $R_1+2\cdot R_3\rightarrow R_1$

Solve the system of equations by using row operations to write a matrix in REDUCED row-echelon form.

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$

Write the system of equations represented by the augmented matrix shown.

Perform each matrix row operation and write the new matrix.

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

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

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.

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

Write the augmented matrix for each system of linear equations.

$\begin{cases}x - y + z = 8 \\y - 12z = -15 \\z = 1\end{cases}$

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

$\{\begin{array}{c}x+2y+3z=-5\\ 2x+y+z=1\\ x+y-z=8\end{array}$