Linear Algebra Examples

$[\begin{array}{c} 10 & 9 \\ - 6 & - 5 \end{array}]$

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

$10 \cdot - 5 - (- 6 \cdot 9)$

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 $10$ by $- 5$ .

$- 50 - (- 6 \cdot 9)$

Step 2.2.1.2

Multiply $- (- 6 \cdot 9)$ .

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Step 2.2.1.2.1

Multiply $- 6$ by $9$ .

$- 50 - - 54$

Step 2.2.1.2.2

Multiply $- 1$ by $- 54$ .

$- 50 + 54$

Step 2.2.2

Add $- 50$ and $54$ .

$4$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{4} [\begin{array}{c} - 5 & - 9 \\ 6 & 10 \end{array}]$

Step 5

Multiply $\frac{1}{4}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Combine $\frac{1}{4}$ and $- 5$ .

$[\begin{array}{c} \frac{- 5}{4} & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{5}{4} & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.3

Combine $\frac{1}{4}$ and $- 9$ .

$[\begin{array}{c} - \frac{5}{4} & \frac{- 9}{4} \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5

Cancel the common factor of $2$ .

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Step 6.5.1

Factor $2$ out of $4$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 (2)} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.2

Factor $2$ out of $6$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 \cdot 2} \cdot (2 \cdot 3) & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.3

Cancel the common factor.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 \cdot 2} \cdot (2 \cdot 3) & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.4

Rewrite the expression.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2} \cdot 3 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.6

Combine $\frac{1}{2}$ and $3$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.7

Cancel the common factor of $2$ .

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Step 6.7.1

Factor $2$ out of $4$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 (2)} \cdot 10 \end{array}]$

Step 6.7.2

Factor $2$ out of $10$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 \cdot 2} \cdot (2 \cdot 5) \end{array}]$

Step 6.7.3

Cancel the common factor.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 \cdot 2} \cdot (2 \cdot 5) \end{array}]$

Step 6.7.4

Rewrite the expression.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2} \cdot 5 \end{array}]$

Step 6.8

Combine $\frac{1}{2}$ and $5$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{5}{2} \end{array}]$