Solve the Matrix Equation [[-2,3],[4,-9]]z=[[-1,6],[17,-30]]

$[\begin{array}{c} - 2 & 3 \\ 4 & - 9 \end{array}] z = [\begin{array}{c} - 1 & 6 \\ 17 & - 30 \end{array}]$

Find the inverse matrix.

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $| A |$ is the determinant of $A$ .

If $A = [\begin{array}{c} a & b \\ c & d \end{array}]$ then $A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$

Find the determinant of $[\begin{array}{c} - 2 & 3 \\ 4 & - 9 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} - 2 & 3 \\ 4 & - 9 \end{array}] = | \begin{array}{c} - 2 & 3 \\ 4 & - 9 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$(- 2) (- 9) - 4 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $- 9$ .

$18 - 4 \cdot 3$

Multiply $- 4$ by $3$ .

$18 - 12$

Subtract $12$ from $18$ .

$6$

Substitute the known values into the formula for the inverse of a matrix.

$\frac{1}{6} [\begin{array}{c} - 9 & - (3) \\ - (4) & - 2 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (3)$ .

$\frac{1}{6} [\begin{array}{c} - 9 & - 3 \\ - (4) & - 2 \end{array}]$

Rearrange $- (4)$ .

$\frac{1}{6} [\begin{array}{c} - 9 & - 3 \\ - 4 & - 2 \end{array}]$

Multiply $\frac{1}{6}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{6} \cdot - 9 & \frac{1}{6} \cdot - 3 \\ \frac{1}{6} \cdot - 4 & \frac{1}{6} \cdot - 2 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{6} \cdot - 9$ .

$[\begin{array}{c} - \frac{3}{2} & \frac{1}{6} \cdot - 3 \\ \frac{1}{6} \cdot - 4 & \frac{1}{6} \cdot - 2 \end{array}]$

Rearrange $\frac{1}{6} \cdot - 3$ .

$[\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ \frac{1}{6} \cdot - 4 & \frac{1}{6} \cdot - 2 \end{array}]$

Rearrange $\frac{1}{6} \cdot - 4$ .

$[\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & \frac{1}{6} \cdot - 2 \end{array}]$

Rearrange $\frac{1}{6} \cdot - 2$ .

$[\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}]$

Assuming that $X$ is the matrix to solve for, multiply the inverse matrix by both sides of the equation.

$[\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}] \cdot ([\begin{array}{c} - 2 & 3 \\ 4 & - 9 \end{array}] x) = [\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}] \cdot [\begin{array}{c} - 1 & 6 \\ 17 & - 30 \end{array}]$

Simplify the left side of the equation.

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} - \frac{3}{2} \cdot - 2 - \frac{1}{2} \cdot 4 & - \frac{3}{2} \cdot 3 - \frac{1}{2} \cdot - 9 \\ - \frac{2}{3} \cdot - 2 - \frac{1}{3} \cdot 4 & - \frac{2}{3} \cdot 3 - \frac{1}{3} \cdot - 9 \end{array}] x = [\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}] \cdot [\begin{array}{c} - 1 & 6 \\ 17 & - 30 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}] \cdot [\begin{array}{c} - 1 & 6 \\ 17 & - 30 \end{array}]$

Multiplying the identity matrix by any matrix $x$ is matrix $x$ .

$x = [\begin{array}{c} - \frac{3}{2} & - \frac{1}{2} \\ - \frac{2}{3} & - \frac{1}{3} \end{array}] \cdot [\begin{array}{c} - 1 & 6 \\ 17 & - 30 \end{array}]$

Simplify the right side of the equation.

Multiply each row in the first matrix by each column in the second matrix.

$x = [\begin{array}{c} - \frac{3}{2} \cdot - 1 - \frac{1}{2} \cdot 17 & - \frac{3}{2} \cdot 6 - \frac{1}{2} \cdot - 30 \\ - \frac{2}{3} \cdot - 1 - \frac{1}{3} \cdot 17 & - \frac{2}{3} \cdot 6 - \frac{1}{3} \cdot - 30 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$x = [\begin{array}{c} - 7 & 6 \\ - 5 & 6 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} - 7 & 6 \\ - 5 & 6 \end{array}]$

Solve the Matrix Equation [[-2,3],[4,-9]]z=[[-1,6],[17,-30]]

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