7. Systems of Equations & Matrices

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 augmented matrix for each system of linear equations.

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

Perform each matrix row operation and write the new matrix.

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

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.

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