Linear Algebra Examples

$[\begin{array}{c} - 2 & 0 \\ - 5 & - 4 \end{array}] + [\begin{array}{c} - 4 & 0 \\ 2 & 2 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} - 2 - 4 & 0 + 0 \\ - 5 + 2 & - 4 + 2 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Subtract $4$ from $- 2$ .

$[\begin{array}{c} - 6 & 0 + 0 \\ - 5 + 2 & - 4 + 2 \end{array}]$

Step 2.2

Add $0$ and $0$ .

$[\begin{array}{c} - 6 & 0 \\ - 5 + 2 & - 4 + 2 \end{array}]$

Step 2.3

Add $- 5$ and $2$ .

$[\begin{array}{c} - 6 & 0 \\ - 3 & - 4 + 2 \end{array}]$

Step 2.4

Add $- 4$ and $2$ .

$[\begin{array}{c} - 6 & 0 \\ - 3 & - 2 \end{array}]$

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

$- 6 \cdot - 2 - (- 3 \cdot 0)$

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 $- 2$ .

$12 - (- 3 \cdot 0)$

Step 4.2.1.2

Multiply $- (- 3 \cdot 0)$ .

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Step 4.2.1.2.1

Multiply $- 3$ by $0$ .

$12 - 0$

Step 4.2.1.2.2

Multiply $- 1$ by $0$ .

$12 + 0$

Step 4.2.2

Add $12$ and $0$ .

$12$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{12} [\begin{array}{c} - 2 & 0 \\ 3 & - 6 \end{array}]$

Step 7

Multiply $\frac{1}{12}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{12} \cdot - 2 & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $12$ .

$[\begin{array}{c} \frac{1}{2 (6)} \cdot - 2 & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.1.2

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{1}{2 \cdot 6} \cdot (2 \cdot - 1) & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2 \cdot 6} \cdot (2 \cdot - 1) & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{6} \cdot - 1 & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.2

Combine $\frac{1}{6}$ and $- 1$ .

$[\begin{array}{c} \frac{- 1}{6} & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.3

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{6} & \frac{1}{12} \cdot 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.4

Multiply $\frac{1}{12}$ by $0$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{12} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.5

Cancel the common factor of $3$ .

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Step 8.5.1

Factor $3$ out of $12$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{3 (4)} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.5.2

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{3 \cdot 4} \cdot 3 & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.5.3

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{1}{12} \cdot - 6 \end{array}]$

Step 8.6

Cancel the common factor of $6$ .

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Step 8.6.1

Factor $6$ out of $12$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{1}{6 (2)} \cdot - 6 \end{array}]$

Step 8.6.2

Factor $6$ out of $- 6$ .

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{1}{6 \cdot 2} \cdot (6 \cdot - 1) \end{array}]$

Step 8.6.3

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{1}{6 \cdot 2} \cdot (6 \cdot - 1) \end{array}]$

Step 8.6.4

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{1}{2} \cdot - 1 \end{array}]$

Step 8.7

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

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & \frac{- 1}{2} \end{array}]$

Step 8.8

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{6} & 0 \\ \frac{1}{4} & - \frac{1}{2} \end{array}]$

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