Linear Algebra Examples

$[\begin{array}{c} 2 & 4 \\ - 2 & - 4 \end{array}] + [\begin{array}{c} 4 & - 1 \\ 3 & 2 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} 2 + 4 & 4 - 1 \\ - 2 + 3 & - 4 + 2 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Add

$2$ and

$4$ .

$[\begin{array}{c} 6 & 4 - 1 \\ - 2 + 3 & - 4 + 2 \end{array}]$

Step 2.2

Subtract

$1$ from

$4$ .

$[\begin{array}{c} 6 & 3 \\ - 2 + 3 & - 4 + 2 \end{array}]$

Step 2.3

Add

$- 2$ and

$3$ .

$[\begin{array}{c} 6 & 3 \\ 1 & - 4 + 2 \end{array}]$

Step 2.4

Add

$- 4$ and

$2$ .

$[\begin{array}{c} 6 & 3 \\ 1 & - 2 \end{array}]$

Step 3

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 4

Find the determinant.

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Step 4.1

The determinant of a

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

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

$6 \cdot - 2 - 1 \cdot 3$

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

$6$ by

$- 2$ .

$- 12 - 1 \cdot 3$

Step 4.2.1.2

Multiply

$- 1$ by

$3$ .

$- 12 - 3$

Step 4.2.2

Subtract

$3$ from

$- 12$ .

$- 15$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{- 15} [\begin{array}{c} - 2 & - 3 \\ - 1 & 6 \end{array}]$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{15} [\begin{array}{c} - 2 & - 3 \\ - 1 & 6 \end{array}]$

Step 8

Multiply

$- \frac{1}{15}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{15} \cdot - 2 & - \frac{1}{15} \cdot - 3 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Multiply

$- \frac{1}{15} \cdot - 2$ .

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Step 9.1.1

Multiply

$- 2$ by

$- 1$ .

$[\begin{array}{c} 2 (\frac{1}{15}) & - \frac{1}{15} \cdot - 3 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.1.2

Combine

$2$ and

$\frac{1}{15}$ .

$[\begin{array}{c} \frac{2}{15} & - \frac{1}{15} \cdot - 3 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.2

Cancel the common factor of

$3$ .

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Step 9.2.1

Move the leading negative in

$- \frac{1}{15}$ into the numerator.

$[\begin{array}{c} \frac{2}{15} & \frac{- 1}{15} \cdot - 3 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.2.2

Factor

$3$ out of

$15$ .

$[\begin{array}{c} \frac{2}{15} & \frac{- 1}{3 (5)} \cdot - 3 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.2.3

Factor

$3$ out of

$- 3$ .

$[\begin{array}{c} \frac{2}{15} & \frac{- 1}{3 \cdot 5} \cdot (3 \cdot - 1) \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.2.4

Cancel the common factor.

$[\begin{array}{c} \frac{2}{15} & \frac{- 1}{3 \cdot 5} \cdot (3 \cdot - 1) \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.2.5

Rewrite the expression.

$[\begin{array}{c} \frac{2}{15} & \frac{- 1}{5} \cdot - 1 \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.3

Combine

$\frac{- 1}{5}$ and

$- 1$ .

$[\begin{array}{c} \frac{2}{15} & \frac{- - 1}{5} \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.4

Multiply

$- 1$ by

$- 1$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ - \frac{1}{15} \cdot - 1 & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.5

Multiply

$- \frac{1}{15} \cdot - 1$ .

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Step 9.5.1

Multiply

$- 1$ by

$- 1$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ 1 (\frac{1}{15}) & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.5.2

Multiply

$\frac{1}{15}$ by

$1$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & - \frac{1}{15} \cdot 6 \end{array}]$

Step 9.6

Cancel the common factor of

$3$ .

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Step 9.6.1

Move the leading negative in

$- \frac{1}{15}$ into the numerator.

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1}{15} \cdot 6 \end{array}]$

Step 9.6.2

Factor

$3$ out of

$15$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1}{3 (5)} \cdot 6 \end{array}]$

Step 9.6.3

Factor

$3$ out of

$6$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1}{3 \cdot 5} \cdot (3 \cdot 2) \end{array}]$

Step 9.6.4

Cancel the common factor.

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1}{3 \cdot 5} \cdot (3 \cdot 2) \end{array}]$

Step 9.6.5

Rewrite the expression.

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1}{5} \cdot 2 \end{array}]$

Step 9.7

Combine

$\frac{- 1}{5}$ and

$2$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 1 \cdot 2}{5} \end{array}]$

Step 9.8

Multiply

$- 1$ by

$2$ .

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & \frac{- 2}{5} \end{array}]$

Step 9.9

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{2}{15} & \frac{1}{5} \\ \frac{1}{15} & - \frac{2}{5} \end{array}]$

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