Solve the Matrix Equation [[-6,0],[7,1]]X=[[-12,-6],[17,9]]

$[\begin{array}{c} - 6 & 0 \\ 7 & 1 \end{array}] X = [\begin{array}{c} - 12 & - 6 \\ 17 & 9 \end{array}]$

Step 1

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 $[\begin{array}{c} - 6 & 0 \\ 7 & 1 \end{array}]$ .

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

$determinant [\begin{array}{c} - 6 & 0 \\ 7 & 1 \end{array}] = | \begin{array}{c} - 6 & 0 \\ 7 & 1 \end{array} |$

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

$(- 6) (1) - 7 \cdot 0$

Simplify the determinant.

Simplify each term.

Multiply $- 6$ by $1$ .

$- 6 - 7 \cdot 0$

Multiply $- 7$ by $0$ .

$- 6 + 0$

Add $- 6$ and $0$ .

$- 6$

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

$\frac{1}{- 6} [\begin{array}{c} 1 & - (0) \\ - (7) & - 6 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (0)$ .

$\frac{1}{- 6} [\begin{array}{c} 1 & 0 \\ - (7) & - 6 \end{array}]$

Rearrange $- (7)$ .

$\frac{1}{- 6} [\begin{array}{c} 1 & 0 \\ - 7 & - 6 \end{array}]$

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

$[\begin{array}{c} \frac{1}{- 6} \cdot 1 & \frac{1}{- 6} \cdot 0 \\ \frac{1}{- 6} \cdot - 7 & \frac{1}{- 6} \cdot - 6 \end{array}]$

Simplify each element in the matrix.

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

$[\begin{array}{c} - \frac{1}{6} & \frac{1}{- 6} \cdot 0 \\ \frac{1}{- 6} \cdot - 7 & \frac{1}{- 6} \cdot - 6 \end{array}]$

Rearrange $\frac{1}{- 6} \cdot 0$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{- 6} \cdot - 7 & \frac{1}{- 6} \cdot - 6 \end{array}]$

Rearrange $\frac{1}{- 6} \cdot - 7$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & \frac{1}{- 6} \cdot - 6 \end{array}]$

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

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}]$

Step 2

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

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}] \cdot ([\begin{array}{c} - 6 & 0 \\ 7 & 1 \end{array}] x) = [\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}] \cdot [\begin{array}{c} - 12 & - 6 \\ 17 & 9 \end{array}]$

Step 3

Simplify the left side of the equation.

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

$[\begin{array}{c} - \frac{1}{6} \cdot - 6 + 0 \cdot 7 & - \frac{1}{6} \cdot 0 + 0 \cdot 1 \\ \frac{7}{6} \cdot - 6 + 1 \cdot 7 & \frac{7}{6} \cdot 0 + 1 \cdot 1 \end{array}] x = [\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}] \cdot [\begin{array}{c} - 12 & - 6 \\ 17 & 9 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}] \cdot [\begin{array}{c} - 12 & - 6 \\ 17 & 9 \end{array}]$

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

$x = [\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{7}{6} & 1 \end{array}] \cdot [\begin{array}{c} - 12 & - 6 \\ 17 & 9 \end{array}]$

Step 4

Simplify the right side of the equation.

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

$x = [\begin{array}{c} - \frac{1}{6} \cdot - 12 + 0 \cdot 17 & - \frac{1}{6} \cdot - 6 + 0 \cdot 9 \\ \frac{7}{6} \cdot - 12 + 1 \cdot 17 & \frac{7}{6} \cdot - 6 + 1 \cdot 9 \end{array}]$

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

$x = [\begin{array}{c} 2 & 1 \\ 3 & 2 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} 2 & 1 \\ 3 & 2 \end{array}]$

Solve the Matrix Equation [[-6,0],[7,1]]X=[[-12,-6],[17,9]]

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