Examples

[\begin{array}{c} - 2 & 0 \\ - 5 & - 4 \end{array}] - [\begin{array}{c} - 4 & 0 \\ 2 & 2 \end{array}]

$[\begin{array}{c} - 2 & 0 \\ - 5 & - 4 \end{array}] - [\begin{array}{c} - 4 & 0 \\ 2 & 2 \end{array}]$

Step 1

Subtract the corresponding elements.

[\begin{array}{c} - 2 + 4 & 0 - 0 \\ - 5 - 2 & - 4 - 2 \end{array}]

$[\begin{array}{c} - 2 + 4 & 0 - 0 \\ - 5 - 2 & - 4 - 2 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Add

- 2

$- 2$ and

4

$4$ .

[\begin{array}{c} 2 & 0 - 0 \\ - 5 - 2 & - 4 - 2 \end{array}]

$[\begin{array}{c} 2 & 0 - 0 \\ - 5 - 2 & - 4 - 2 \end{array}]$

Step 2.2

Subtract

0

$0$ from

0

$0$ .

[\begin{array}{c} 2 & 0 \\ - 5 - 2 & - 4 - 2 \end{array}]

$[\begin{array}{c} 2 & 0 \\ - 5 - 2 & - 4 - 2 \end{array}]$

Step 2.3

Subtract

2

$2$ from

- 5

$- 5$ .

[\begin{array}{c} 2 & 0 \\ - 7 & - 4 - 2 \end{array}]

$[\begin{array}{c} 2 & 0 \\ - 7 & - 4 - 2 \end{array}]$

Step 2.4

Subtract

2

$2$ from

- 4

$- 4$ .

[\begin{array}{c} 2 & 0 \\ - 7 & - 6 \end{array}]

$[\begin{array}{c} 2 & 0 \\ - 7 & - 6 \end{array}]$

[\begin{array}{c} 2 & 0 \\ - 7 & - 6 \end{array}]

$[\begin{array}{c} 2 & 0 \\ - 7 & - 6 \end{array}]$

Step 3

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 4

Find the determinant.

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Step 4.1

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

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

2 \cdot - 6 - (- 7 \cdot 0)

$2 \cdot - 6 - (- 7 \cdot 0)$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply

2

$2$ by

- 6

$- 6$ .

- 12 - (- 7 \cdot 0)

$- 12 - (- 7 \cdot 0)$

Step 4.2.1.2

Multiply

- (- 7 \cdot 0)

$- (- 7 \cdot 0)$ .

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Step 4.2.1.2.1

Multiply

- 7

$- 7$ by

0

$0$ .

- 12 - 0

$- 12 - 0$

Step 4.2.1.2.2

Multiply

- 1

$- 1$ by

0

$0$ .

- 12 + 0

$- 12 + 0$

- 12 + 0

$- 12 + 0$

- 12 + 0

$- 12 + 0$

Step 4.2.2

Add

- 12

$- 12$ and

0

$0$ .

- 12

$- 12$

- 12

$- 12$

- 12

$- 12$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

\frac{1}{- 12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]

$\frac{1}{- 12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]$

Step 7

Move the negative in front of the fraction.

- \frac{1}{12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]

$- \frac{1}{12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]$

Step 8

Multiply

- \frac{1}{12}

$- \frac{1}{12}$ by each element of the matrix.

[\begin{array}{c} - \frac{1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} - \frac{1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of

6

$6$ .

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Step 9.1.1

Move the leading negative in

- \frac{1}{12}

$- \frac{1}{12}$ into the numerator.

[\begin{array}{c} \frac{- 1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.2

Factor

6

$6$ out of

12

$12$ .

[\begin{array}{c} \frac{- 1}{6 (2)} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{6 (2)} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.3

Factor

6

$6$ out of

- 6

$- 6$ .

[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.4

Cancel the common factor.

[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.5

Rewrite the expression.

[\begin{array}{c} \frac{- 1}{2} \cdot - 1 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{2} \cdot - 1 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

[\begin{array}{c} \frac{- 1}{2} \cdot - 1 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- 1}{2} \cdot - 1 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.2

Combine

\frac{- 1}{2}

$\frac{- 1}{2}$ and

- 1

$- 1$ .

[\begin{array}{c} \frac{- - 1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{- - 1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.3

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

[\begin{array}{c} \frac{1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.4

Multiply

- \frac{1}{12} \cdot 0

$- \frac{1}{12} \cdot 0$ .

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Step 9.4.1

Multiply

0

$0$ by

- 1

$- 1$ .

[\begin{array}{c} \frac{1}{2} & 0 (\frac{1}{12}) \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 (\frac{1}{12}) \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.4.2

Multiply

0

$0$ by

\frac{1}{12}

$\frac{1}{12}$ .

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.5

Multiply

- \frac{1}{12} \cdot 7

$- \frac{1}{12} \cdot 7$ .

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Step 9.5.1

Multiply

7

$7$ by

- 1

$- 1$ .

[\begin{array}{c} \frac{1}{2} & 0 \\ - 7 (\frac{1}{12}) & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - 7 (\frac{1}{12}) & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.5.2

Combine

- 7

$- 7$ and

\frac{1}{12}

$\frac{1}{12}$ .

[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{- 7}{12} & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{- 7}{12} & - \frac{1}{12} \cdot 2 \end{array}]$

[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{- 7}{12} & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{- 7}{12} & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.6

Move the negative in front of the fraction.

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.7

Cancel the common factor of

2

$2$ .

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Step 9.7.1

Move the leading negative in

- \frac{1}{12}

$- \frac{1}{12}$ into the numerator.

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{12} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{12} \cdot 2 \end{array}]$

Step 9.7.2

Factor

2

$2$ out of

12

$12$ .

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 (6)} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 (6)} \cdot 2 \end{array}]$

Step 9.7.3

Cancel the common factor.

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 \cdot 6} \cdot 2 \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 \cdot 6} \cdot 2 \end{array}]$

Step 9.7.4

Rewrite the expression.

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{6} \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{6} \end{array}]$

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{6} \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{6} \end{array}]$

Step 9.8

Move the negative in front of the fraction.

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{6} \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{6} \end{array}]$

[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{6} \end{array}]

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{6} \end{array}]$

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