Linear Algebra Examples

$[\begin{array}{c} 1 & 7 \\ 4 & 2 \end{array}]$

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

$1 \cdot 2 - 4 \cdot 7$

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 $2$ by $1$ .

$2 - 4 \cdot 7$

Step 2.2.1.2

Multiply $- 4$ by $7$ .

$2 - 28$

Step 2.2.2

Subtract $28$ from $2$ .

$- 26$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 26} [\begin{array}{c} 2 & - 7 \\ - 4 & 1 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{26} [\begin{array}{c} 2 & - 7 \\ - 4 & 1 \end{array}]$

Step 6

Multiply $- \frac{1}{26}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{26} \cdot 2 & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Move the leading negative in $- \frac{1}{26}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{26} \cdot 2 & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.1.2

Factor $2$ out of $26$ .

$[\begin{array}{c} \frac{- 1}{2 (13)} \cdot 2 & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 13} \cdot 2 & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{13} & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{13} & - \frac{1}{26} \cdot - 7 \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.3

Multiply $- \frac{1}{26} \cdot - 7$ .

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Step 7.3.1

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{13} & 7 (\frac{1}{26}) \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.3.2

Combine $7$ and $\frac{1}{26}$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ - \frac{1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Move the leading negative in $- \frac{1}{26}$ into the numerator.

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- 1}{26} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.4.2

Factor $2$ out of $26$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- 1}{2 (13)} \cdot - 4 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.4.3

Factor $2$ out of $- 4$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- 1}{2 \cdot 13} \cdot (2 \cdot - 2) & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.4.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- 1}{2 \cdot 13} \cdot (2 \cdot - 2) & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.4.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- 1}{13} \cdot - 2 & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.5

Combine $\frac{- 1}{13}$ and $- 2$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{- - 2}{13} & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.6

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{2}{13} & - \frac{1}{26} \cdot 1 \end{array}]$

Step 7.7

Multiply $- 1$ by $1$ .

$[\begin{array}{c} - \frac{1}{13} & \frac{7}{26} \\ \frac{2}{13} & - \frac{1}{26} \end{array}]$