Find the Inverse [[8,-4],[-8,4]]

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

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 $[\begin{array}{c} 8 & - 4 \\ - 8 & 4 \end{array}]$ .

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

$determinant [\begin{array}{c} 8 & - 4 \\ - 8 & 4 \end{array}] = | \begin{array}{c} 8 & - 4 \\ - 8 & 4 \end{array} |$

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

$(8) (4) + 8 \cdot - 4$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $4$ .

$32 + 8 \cdot - 4$

Multiply $8$ by $- 4$ .

$32 - 32$

Subtract $32$ from $32$ .

$0$

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

$\frac{1}{0} [\begin{array}{c} 4 & - (- 4) \\ - (- 8) & 8 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 4)$ .

$\frac{1}{0} [\begin{array}{c} 4 & 4 \\ - (- 8) & 8 \end{array}]$

Rearrange $- (- 8)$ .

$\frac{1}{0} [\begin{array}{c} 4 & 4 \\ 8 & 8 \end{array}]$

Multiply $\frac{1}{0}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{0} \cdot 4 & \frac{1}{0} \cdot 4 \\ \frac{1}{0} \cdot 8 & \frac{1}{0} \cdot 8 \end{array}]$

Rearrange $\frac{1}{0} \cdot 4$ .

$[\begin{array}{c} Undefined & \frac{1}{0} \cdot 4 \\ \frac{1}{0} \cdot 8 & \frac{1}{0} \cdot 8 \end{array}]$

Since the matrix is undefined, it cannot be solved.

Undefined

Find the Inverse [[8,-4],[-8,4]]

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