Linear Algebra Examples

[\begin{matrix} 2 & 1 4 & 1 \end{matrix}] - [\begin{matrix} - 3 & - 3 - 5 & - 3 \end{matrix}]

$[\begin{array}{c} 2 & 1 \\ 4 & 1 \end{array}] - [\begin{array}{c} - 3 & - 3 \\ - 5 & - 3 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} 2 + 3 & 1 + 3 \\ 4 + 5 & 1 + 3 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Add

$2$ and

$3$ .

$[\begin{array}{c} 5 & 1 + 3 \\ 4 + 5 & 1 + 3 \end{array}]$

Step 2.2

Add

$1$ and

$3$ .

$[\begin{array}{c} 5 & 4 \\ 4 + 5 & 1 + 3 \end{array}]$

Step 2.3

Add

$4$ and

$5$ .

$[\begin{array}{c} 5 & 4 \\ 9 & 1 + 3 \end{array}]$

Step 2.4

Add

$1$ and

$3$ .

$[\begin{array}{c} 5 & 4 \\ 9 & 4 \end{array}]$

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

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

$5 \cdot 4 - 9 \cdot 4$

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

$5$ by

$4$ .

$20 - 9 \cdot 4$

Step 4.2.1.2

Multiply

$- 9$ by

$4$ .

$20 - 36$

Step 4.2.2

Subtract

$36$ from

$20$ .

$- 16$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{- 16} [\begin{array}{c} 4 & - 4 \\ - 9 & 5 \end{array}]$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{16} [\begin{array}{c} 4 & - 4 \\ - 9 & 5 \end{array}]$

Step 8

Multiply

$- \frac{1}{16}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{16} \cdot 4 & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of

$4$ .

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Step 9.1.1

Move the leading negative in

$- \frac{1}{16}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{16} \cdot 4 & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.1.2

Factor

$4$ out of

$16$ .

$[\begin{array}{c} \frac{- 1}{4 (4)} \cdot 4 & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{4 \cdot 4} \cdot 4 & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{4} & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{4} & - \frac{1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.3

Cancel the common factor of

$4$ .

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Step 9.3.1

Move the leading negative in

$- \frac{1}{16}$ into the numerator.

$[\begin{array}{c} - \frac{1}{4} & \frac{- 1}{16} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.3.2

Factor

$4$ out of

$16$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{- 1}{4 (4)} \cdot - 4 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.3.3

Factor

$4$ out of

$- 4$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{- 1}{4 \cdot 4} \cdot (4 \cdot - 1) \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.3.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{4} & \frac{- 1}{4 \cdot 4} \cdot (4 \cdot - 1) \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.3.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{4} & \frac{- 1}{4} \cdot - 1 \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.4

Combine

$\frac{- 1}{4}$ and

$- 1$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{- - 1}{4} \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.5

Multiply

$- 1$ by

$- 1$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ - \frac{1}{16} \cdot - 9 & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.6

Multiply

$- \frac{1}{16} \cdot - 9$ .

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Step 9.6.1

Multiply

$- 9$ by

$- 1$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ 9 (\frac{1}{16}) & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.6.2

Combine

$9$ and

$\frac{1}{16}$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ \frac{9}{16} & - \frac{1}{16} \cdot 5 \end{array}]$

Step 9.7

Multiply

$- \frac{1}{16} \cdot 5$ .

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Step 9.7.1

Multiply

$5$ by

$- 1$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ \frac{9}{16} & - 5 (\frac{1}{16}) \end{array}]$

Step 9.7.2

Combine

$- 5$ and

$\frac{1}{16}$ .

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ \frac{9}{16} & \frac{- 5}{16} \end{array}]$

Step 9.8

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{4} & \frac{1}{4} \\ \frac{9}{16} & - \frac{5}{16} \end{array}]$

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