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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