Examples

[\begin{array}{c} 1 & 2 \\ 3 & 5 \end{array}]

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

Step 1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Step 2

Find the determinant.

Tap for more steps...

Step 2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

1 \cdot 5 - 3 \cdot 2

$1 \cdot 5 - 3 \cdot 2$

Step 2.2

Simplify the determinant.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply

5

$5$ by

1

$1$ .

5 - 3 \cdot 2

$5 - 3 \cdot 2$

Step 2.2.1.2

Multiply

- 3

$- 3$ by

2

$2$ .

5 - 6

$5 - 6$

5 - 6

$5 - 6$

Step 2.2.2

Subtract

6

$6$ from

5

$5$ .

- 1

$- 1$

- 1

$- 1$

- 1

$- 1$

Step 3

Since the determinant is non-zero, the inverse exists.

Step 4

Substitute the known values into the formula for the inverse.

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

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

Step 5

Divide

1

$1$ by

- 1

$- 1$ .

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

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

Step 6

Multiply

- 1

$- 1$ by each element of the matrix.

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

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

Step 7

Simplify each element in the matrix.

Tap for more steps...

Step 7.1

Multiply

- 1

$- 1$ by

5

$5$ .

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

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

Step 7.2

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

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

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

Step 7.3

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

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

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

Step 7.4

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

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

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

Enter YOUR Problem