Linear Algebra Examples

$[\begin{array}{c} 1 & - 7 \\ - 8 & 0 \end{array}]$

Step 1

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

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

$1 \cdot 0 - (- 8 \cdot - 7)$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $0$ by $1$ .

$0 - (- 8 \cdot - 7)$

Step 2.2.1.2

Multiply $- (- 8 \cdot - 7)$ .

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Step 2.2.1.2.1

Multiply $- 8$ by $- 7$ .

$0 - 1 \cdot 56$

Step 2.2.1.2.2

Multiply $- 1$ by $56$ .

$0 - 56$

Step 2.2.2

Subtract $56$ from $0$ .

$- 56$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 56} [\begin{array}{c} 0 & 7 \\ 8 & 1 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{56} [\begin{array}{c} 0 & 7 \\ 8 & 1 \end{array}]$

Step 6

Multiply $- \frac{1}{56}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{56} \cdot 0 & - \frac{1}{56} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- \frac{1}{56} \cdot 0$ .

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Step 7.1.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 0 (\frac{1}{56}) & - \frac{1}{56} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.1.2

Multiply $0$ by $\frac{1}{56}$ .

$[\begin{array}{c} 0 & - \frac{1}{56} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.2

Cancel the common factor of $7$ .

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Step 7.2.1

Move the leading negative in $- \frac{1}{56}$ into the numerator.

$[\begin{array}{c} 0 & \frac{- 1}{56} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.2.2

Factor $7$ out of $56$ .

$[\begin{array}{c} 0 & \frac{- 1}{7 (8)} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.2.3

Cancel the common factor.

$[\begin{array}{c} 0 & \frac{- 1}{7 \cdot 8} \cdot 7 \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.2.4

Rewrite the expression.

$[\begin{array}{c} 0 & \frac{- 1}{8} \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.3

Move the negative in front of the fraction.

$[\begin{array}{c} 0 & - \frac{1}{8} \\ - \frac{1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.4

Cancel the common factor of $8$ .

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Step 7.4.1

Move the leading negative in $- \frac{1}{56}$ into the numerator.

$[\begin{array}{c} 0 & - \frac{1}{8} \\ \frac{- 1}{56} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.4.2

Factor $8$ out of $56$ .

$[\begin{array}{c} 0 & - \frac{1}{8} \\ \frac{- 1}{8 (7)} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.4.3

Cancel the common factor.

$[\begin{array}{c} 0 & - \frac{1}{8} \\ \frac{- 1}{8 \cdot 7} \cdot 8 & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.4.4

Rewrite the expression.

$[\begin{array}{c} 0 & - \frac{1}{8} \\ \frac{- 1}{7} & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.5

Move the negative in front of the fraction.

$[\begin{array}{c} 0 & - \frac{1}{8} \\ - \frac{1}{7} & - \frac{1}{56} \cdot 1 \end{array}]$

Step 7.6

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 0 & - \frac{1}{8} \\ - \frac{1}{7} & - \frac{1}{56} \end{array}]$