Linear Algebra Examples

[\begin{matrix} 3 & - 9 - 2 & 5 \end{matrix}]

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

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{matrix} d & - b - c & a \end{matrix}]

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

a d - b c

$a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

The determinant of a

2 \times 2

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

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

3 \cdot 5 - (- 2 \cdot - 9)

$3 \cdot 5 - (- 2 \cdot - 9)$

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

3

$3$ by

5

$5$ .

15 - (- 2 \cdot - 9)

$15 - (- 2 \cdot - 9)$

Step 2.2.1.2

Multiply

- (- 2 \cdot - 9)

$- (- 2 \cdot - 9)$ .

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Step 2.2.1.2.1

Multiply

- 2

$- 2$ by

- 9

$- 9$ .

15 - 1 \cdot 18

$15 - 1 \cdot 18$

Step 2.2.1.2.2

Multiply

- 1

$- 1$ by

18

$18$ .

15 - 18

$15 - 18$

15 - 18

$15 - 18$

15 - 18

$15 - 18$

Step 2.2.2

Subtract

18

$18$ from

15

$15$ .

- 3

$- 3$

- 3

$- 3$

- 3

$- 3$

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}{- 3} [\begin{matrix} 5 & 9 2 & 3 \end{matrix}]

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

Step 5

Move the negative in front of the fraction.

- \frac{1}{3} [\begin{matrix} 5 & 9 2 & 3 \end{matrix}]

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

Step 6

Multiply

- \frac{1}{3}

$- \frac{1}{3}$ by each element of the matrix.

[\begin{matrix} - \frac{1}{3} \cdot 5 & - \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply

- \frac{1}{3} \cdot 5

$- \frac{1}{3} \cdot 5$ .

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Step 7.1.1

Multiply

5

$5$ by

- 1

$- 1$ .

⎡ ⎢ ⎣ \begin{matrix} - 5 (\frac{1}{3}) & - \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix} ⎤ ⎥ ⎦

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

Step 7.1.2

Combine

- 5

$- 5$ and

\frac{1}{3}

$\frac{1}{3}$ .

[\begin{matrix} \frac{- 5}{3} & - \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

[\begin{matrix} \frac{- 5}{3} & - \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

Step 7.2

Move the negative in front of the fraction.

[\begin{matrix} - \frac{5}{3} & - \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

Step 7.3

Cancel the common factor of

3

$3$ .

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Step 7.3.1

Move the leading negative in

- \frac{1}{3}

$- \frac{1}{3}$ into the numerator.

[\begin{matrix} - \frac{5}{3} & \frac{- 1}{3} \cdot 9 - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

Step 7.3.2

Factor

3

$3$ out of

9

$9$ .

[\begin{matrix} - \frac{5}{3} & \frac{- 1}{3} \cdot (3 (3)) - \frac{1}{3} \cdot 2 & - \frac{1}{3} \cdot 3 \end{matrix}]

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

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

Step 7.4

Multiply

$- 1$ by

$3$ .

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

Step 7.5

Multiply

$- \frac{1}{3} \cdot 2$ .

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Step 7.5.1

Multiply

$2$ by

$- 1$ .

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

Step 7.5.2

Combine

$- 2$ and

$\frac{1}{3}$ .

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

Step 7.6

Move the negative in front of the fraction.

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

Step 7.7

Cancel the common factor of

$3$ .

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Step 7.7.1

Move the leading negative in

$- \frac{1}{3}$ into the numerator.

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

Step 7.7.2

Cancel the common factor.

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

Step 7.7.3

Rewrite the expression.

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