Linear Algebra Examples

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

Step 1

Subtract 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

Add $- 2$ and $4$ .

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

Step 2.2

Subtract $0$ from $0$ .

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

Step 2.3

Subtract $2$ from $- 5$ .

$[\begin{array}{c} 2 & 0 \\ - 7 & - 4 - 2 \end{array}]$

Step 2.4

Subtract $2$ from $- 4$ .

$[\begin{array}{c} 2 & 0 \\ - 7 & - 6 \end{array}]$

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

$2 \cdot - 6 - (- 7 \cdot 0)$

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 $2$ by $- 6$ .

$- 12 - (- 7 \cdot 0)$

Step 4.2.1.2

Multiply $- (- 7 \cdot 0)$ .

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Step 4.2.1.2.1

Multiply $- 7$ 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.

$\frac{1}{- 12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{12} [\begin{array}{c} - 6 & 0 \\ 7 & 2 \end{array}]$

Step 8

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

$[\begin{array}{c} - \frac{1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of $6$ .

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Step 9.1.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{12} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.2

Factor $6$ out of $12$ .

$[\begin{array}{c} \frac{- 1}{6 (2)} \cdot - 6 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.3

Factor $6$ out of $- 6$ .

$[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.4

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{6 \cdot 2} \cdot (6 \cdot - 1) & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.1.5

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{2} \cdot - 1 & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.2

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

$[\begin{array}{c} \frac{- - 1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.3

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{12} \cdot 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.4

Multiply $- \frac{1}{12} \cdot 0$ .

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Step 9.4.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} \frac{1}{2} & 0 (\frac{1}{12}) \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.4.2

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

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{1}{12} \cdot 7 & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.5

Multiply $- \frac{1}{12} \cdot 7$ .

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Step 9.5.1

Multiply $7$ by $- 1$ .

$[\begin{array}{c} \frac{1}{2} & 0 \\ - 7 (\frac{1}{12}) & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.5.2

Combine $- 7$ and $\frac{1}{12}$ .

$[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{- 7}{12} & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{12} \cdot 2 \end{array}]$

Step 9.7

Cancel the common factor of $2$ .

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Step 9.7.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{12} \cdot 2 \end{array}]$

Step 9.7.2

Factor $2$ out of $12$ .

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 (6)} \cdot 2 \end{array}]$

Step 9.7.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{2 \cdot 6} \cdot 2 \end{array}]$

Step 9.7.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & \frac{- 1}{6} \end{array}]$

Step 9.8

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{2} & 0 \\ - \frac{7}{12} & - \frac{1}{6} \end{array}]$

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