Find the Inverse [[6,6],[3,-1]]

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

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 $[\begin{array}{c} 6 & 6 \\ 3 & - 1 \end{array}]$ .

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

$determinant [\begin{array}{c} 6 & 6 \\ 3 & - 1 \end{array}] = | \begin{array}{c} 6 & 6 \\ 3 & - 1 \end{array} |$

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

$(6) (- 1) - 3 \cdot 6$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $- 1$ .

$- 6 - 3 \cdot 6$

Multiply $- 3$ by $6$ .

$- 6 - 18$

Subtract $18$ from $- 6$ .

$- 24$

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

$\frac{1}{- 24} [\begin{array}{c} - 1 & - (6) \\ - (3) & 6 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (6)$ .

$\frac{1}{- 24} [\begin{array}{c} - 1 & - 6 \\ - (3) & 6 \end{array}]$

Rearrange $- (3)$ .

$\frac{1}{- 24} [\begin{array}{c} - 1 & - 6 \\ - 3 & 6 \end{array}]$

Multiply $\frac{1}{- 24}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{- 24} \cdot - 1 & \frac{1}{- 24} \cdot - 6 \\ \frac{1}{- 24} \cdot - 3 & \frac{1}{- 24} \cdot 6 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{- 24} \cdot - 1$ .

$[\begin{array}{c} \frac{1}{24} & \frac{1}{- 24} \cdot - 6 \\ \frac{1}{- 24} \cdot - 3 & \frac{1}{- 24} \cdot 6 \end{array}]$

Rearrange $\frac{1}{- 24} \cdot - 6$ .

$[\begin{array}{c} \frac{1}{24} & \frac{1}{4} \\ \frac{1}{- 24} \cdot - 3 & \frac{1}{- 24} \cdot 6 \end{array}]$

Rearrange $\frac{1}{- 24} \cdot - 3$ .

$[\begin{array}{c} \frac{1}{24} & \frac{1}{4} \\ \frac{1}{8} & \frac{1}{- 24} \cdot 6 \end{array}]$

Rearrange $\frac{1}{- 24} \cdot 6$ .

$[\begin{array}{c} \frac{1}{24} & \frac{1}{4} \\ \frac{1}{8} & - \frac{1}{4} \end{array}]$

Find the Inverse [[6,6],[3,-1]]

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