Find the Inverse A=[[3,2],[8,5]]

$A = [\begin{array}{c} 3 & 2 \\ 8 & 5 \end{array}]$

Step 1

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

Step 2

Find the determinant of $[\begin{array}{c} 3 & 2 \\ 8 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 3 & 2 \\ 8 & 5 \end{array}] = | \begin{array}{c} 3 & 2 \\ 8 & 5 \end{array} |$

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

$(3) (5) - 8 \cdot 2$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $5$ .

$15 - 8 \cdot 2$

Multiply $- 8$ by $2$ .

$15 - 16$

Subtract $16$ from $15$ .

$- 1$

Step 3

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

$\frac{1}{- 1} [\begin{array}{c} 5 & - (2) \\ - (8) & 3 \end{array}]$

Step 4

Simplify each element in the matrix.

Rearrange $- (2)$ .

$\frac{1}{- 1} [\begin{array}{c} 5 & - 2 \\ - (8) & 3 \end{array}]$

Rearrange $- (8)$ .

$\frac{1}{- 1} [\begin{array}{c} 5 & - 2 \\ - 8 & 3 \end{array}]$

Step 5

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

$[\begin{array}{c} \frac{1}{- 1} \cdot 5 & \frac{1}{- 1} \cdot - 2 \\ \frac{1}{- 1} \cdot - 8 & \frac{1}{- 1} \cdot 3 \end{array}]$

Step 6

Simplify each element in the matrix.

Rearrange $\frac{1}{- 1} \cdot 5$ .

$[\begin{array}{c} - 5 & \frac{1}{- 1} \cdot - 2 \\ \frac{1}{- 1} \cdot - 8 & \frac{1}{- 1} \cdot 3 \end{array}]$

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

$[\begin{array}{c} - 5 & 2 \\ \frac{1}{- 1} \cdot - 8 & \frac{1}{- 1} \cdot 3 \end{array}]$

Rearrange $\frac{1}{- 1} \cdot - 8$ .

$[\begin{array}{c} - 5 & 2 \\ 8 & \frac{1}{- 1} \cdot 3 \end{array}]$

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

$[\begin{array}{c} - 5 & 2 \\ 8 & - 3 \end{array}]$

Find the Inverse A=[[3,2],[8,5]]

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