Find the Inverse [[10,-5],[-4,2]]

$[\begin{array}{c} 10 & - 5 \\ - 4 & 2 \end{array}]$

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 $[\begin{array}{c} 10 & - 5 \\ - 4 & 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 10 & - 5 \\ - 4 & 2 \end{array}] = | \begin{array}{c} 10 & - 5 \\ - 4 & 2 \end{array} |$

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

$(10) (2) + 4 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $2$ .

$20 + 4 \cdot - 5$

Multiply $4$ by $- 5$ .

$20 - 20$

Subtract $20$ from $20$ .

$0$

Step 3

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

$\frac{1}{0} [\begin{array}{c} 2 & - (- 5) \\ - (- 4) & 10 \end{array}]$

Step 4

Simplify each element in the matrix.

Rearrange $- (- 5)$ .

$\frac{1}{0} [\begin{array}{c} 2 & 5 \\ - (- 4) & 10 \end{array}]$

Rearrange $- (- 4)$ .

$\frac{1}{0} [\begin{array}{c} 2 & 5 \\ 4 & 10 \end{array}]$

Step 5

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

$[\begin{array}{c} \frac{1}{0} \cdot 2 & \frac{1}{0} \cdot 5 \\ \frac{1}{0} \cdot 4 & \frac{1}{0} \cdot 10 \end{array}]$

Step 6

Rearrange $\frac{1}{0} \cdot 2$ .

$[\begin{array}{c} Undefined & \frac{1}{0} \cdot 5 \\ \frac{1}{0} \cdot 4 & \frac{1}{0} \cdot 10 \end{array}]$

Step 7

Since the matrix is undefined, it cannot be solved.

Undefined

Find the Inverse [[10,-5],[-4,2]]

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