Linear Algebra Examples

$[\begin{array}{c} - 6 x + 7 y & - 25 \\ - 4 x - 5 y & - 7 \end{array}]$

Step 1

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 2

Find the determinant.

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Step 2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$(- 6 x + 7 y) \cdot - 7 - (- 4 x - 5 y) \cdot - 25$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$- 6 x \cdot - 7 + 7 y \cdot - 7 - (- 4 x - 5 y) \cdot - 25$

Step 2.2.1.2

Multiply $- 7$ by $- 6$ .

$42 x + 7 y \cdot - 7 - (- 4 x - 5 y) \cdot - 25$

Step 2.2.1.3

Multiply $- 7$ by $7$ .

$42 x - 49 y - (- 4 x - 5 y) \cdot - 25$

Step 2.2.1.4

Apply the distributive property.

$42 x - 49 y + (- (- 4 x) - (- 5 y)) \cdot - 25$

Step 2.2.1.5

Multiply $- 4$ by $- 1$ .

$42 x - 49 y + (4 x - (- 5 y)) \cdot - 25$

Step 2.2.1.6

Multiply $- 5$ by $- 1$ .

$42 x - 49 y + (4 x + 5 y) \cdot - 25$

Step 2.2.1.7

Apply the distributive property.

$42 x - 49 y + 4 x \cdot - 25 + 5 y \cdot - 25$

Step 2.2.1.8

Multiply $- 25$ by $4$ .

$42 x - 49 y - 100 x + 5 y \cdot - 25$

Step 2.2.1.9

Multiply $- 25$ by $5$ .

$42 x - 49 y - 100 x - 125 y$

Step 2.2.2

Subtract $100 x$ from $42 x$ .

$- 58 x - 49 y - 125 y$

Step 2.2.3

Subtract $125 y$ from $- 49 y$ .

$- 58 x - 174 y$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 58 x - 174 y} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 5

Factor $58$ out of $- 58 x - 174 y$ .

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Step 5.1

Factor $58$ out of $- 58 x$ .

$\frac{1}{58 (- x) - 174 y} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 5.2

Factor $58$ out of $- 174 y$ .

$\frac{1}{58 (- x) + 58 (- 3 y)} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 5.3

Factor $58$ out of $58 (- x) + 58 (- 3 y)$ .

$\frac{1}{58 (- x - 3 y)} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 6

Factor $- 1$ out of $- x$ .

$\frac{1}{58 (- (x) - 3 y)} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 7

Factor $- 1$ out of $- 3 y$ .

$\frac{1}{58 (- (x) - (3 y))} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 8

Factor $- 1$ out of $- (x) - (3 y)$ .

$\frac{1}{58 (- (x + 3 y))} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 9

Rewrite $- (x + 3 y)$ as $- 1 (x + 3 y)$ .

$\frac{1}{58 (- 1 (x + 3 y))} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 10

Move the negative in front of the fraction.

$- \frac{1}{58 (x + 3 y)} [\begin{array}{c} - 7 & 25 \\ - (- 4 x - 5 y) & - 6 x + 7 y \end{array}]$

Step 11

Apply the distributive property.

$- \frac{1}{58 (x + 3 y)} [\begin{array}{c} - 7 & 25 \\ - (- 4 x) - (- 5 y) & - 6 x + 7 y \end{array}]$

Step 12

Multiply $- 4$ by $- 1$ .

$- \frac{1}{58 (x + 3 y)} [\begin{array}{c} - 7 & 25 \\ 4 x - (- 5 y) & - 6 x + 7 y \end{array}]$

Step 13

Multiply $- 5$ by $- 1$ .

$- \frac{1}{58 (x + 3 y)} [\begin{array}{c} - 7 & 25 \\ 4 x + 5 y & - 6 x + 7 y \end{array}]$

Step 14

Multiply $- \frac{1}{58 (x + 3 y)}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{58 (x + 3 y)} \cdot - 7 & - \frac{1}{58 (x + 3 y)} \cdot 25 \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15

Simplify each element in the matrix.

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Step 15.1

Multiply $- \frac{1}{58 (x + 3 y)} \cdot - 7$ .

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Step 15.1.1

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} 7 \frac{1}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} \cdot 25 \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.1.2

Combine $7$ and $\frac{1}{58 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} \cdot 25 \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.2

Multiply $- \frac{1}{58 (x + 3 y)} \cdot 25$ .

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Step 15.2.1

Multiply $25$ by $- 1$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - 25 \frac{1}{58 (x + 3 y)} \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.2.2

Combine $- 25$ and $\frac{1}{58 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & \frac{- 25}{58 (x + 3 y)} \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.3

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{1}{58 (x + 3 y)} (4 x + 5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.4

Apply the distributive property.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{1}{58 (x + 3 y)} (4 x) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.5

Cancel the common factor of $2$ .

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Step 15.5.1

Move the leading negative in $- \frac{1}{58 (x + 3 y)}$ into the numerator.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 1}{58 (x + 3 y)} (4 x) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.5.2

Factor $2$ out of $58 (x + 3 y)$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 1}{2 (29 (x + 3 y))} (4 x) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.5.3

Factor $2$ out of $4 x$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 1}{2 (29 (x + 3 y))} (2 (2 x)) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.5.4

Cancel the common factor.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 1}{2 (29 (x + 3 y))} (2 (2 x)) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.5.5

Rewrite the expression.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 1}{29 (x + 3 y)} (2 x) - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.6

Combine $2$ and $\frac{- 1}{29 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{2 \cdot - 1}{29 (x + 3 y)} x - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.7

Multiply $2$ by $- 1$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 2}{29 (x + 3 y)} x - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.8

Combine $\frac{- 2}{29 (x + 3 y)}$ and $x$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 2 x}{29 (x + 3 y)} - \frac{1}{58 (x + 3 y)} (5 y) & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.9

Multiply $- \frac{1}{58 (x + 3 y)} (5 y)$ .

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Step 15.9.1

Multiply $5$ by $- 1$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 2 x}{29 (x + 3 y)} - 5 \frac{1}{58 (x + 3 y)} y & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.9.2

Combine $- 5$ and $\frac{1}{58 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 2 x}{29 (x + 3 y)} + \frac{- 5}{58 (x + 3 y)} y & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.9.3

Combine $\frac{- 5}{58 (x + 3 y)}$ and $y$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ \frac{- 2 x}{29 (x + 3 y)} + \frac{- 5 y}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.10

Simplify each term.

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Step 15.10.1

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} + \frac{- 5 y}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.10.2

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} (- 6 x + 7 y) \end{array}]$

Step 15.11

Apply the distributive property.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & - \frac{1}{58 (x + 3 y)} (- 6 x) - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.12

Cancel the common factor of $2$ .

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Step 15.12.1

Move the leading negative in $- \frac{1}{58 (x + 3 y)}$ into the numerator.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{- 1}{58 (x + 3 y)} (- 6 x) - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.12.2

Factor $2$ out of $58 (x + 3 y)$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{- 1}{2 (29 (x + 3 y))} (- 6 x) - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.12.3

Factor $2$ out of $- 6 x$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{- 1}{2 (29 (x + 3 y))} (2 (- 3 x)) - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.12.4

Cancel the common factor.

Step 15.12.5

Rewrite the expression.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{- 1}{29 (x + 3 y)} (- 3 x) - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.13

Combine $- 3$ and $\frac{- 1}{29 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{- 3 \cdot - 1}{29 (x + 3 y)} x - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.14

Multiply $- 3$ by $- 1$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3}{29 (x + 3 y)} x - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.15

Combine $\frac{3}{29 (x + 3 y)}$ and $x$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3 x}{29 (x + 3 y)} - \frac{1}{58 (x + 3 y)} (7 y) \end{array}]$

Step 15.16

Multiply $- \frac{1}{58 (x + 3 y)} (7 y)$ .

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Step 15.16.1

Multiply $7$ by $- 1$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3 x}{29 (x + 3 y)} - 7 \frac{1}{58 (x + 3 y)} y \end{array}]$

Step 15.16.2

Combine $- 7$ and $\frac{1}{58 (x + 3 y)}$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3 x}{29 (x + 3 y)} + \frac{- 7}{58 (x + 3 y)} y \end{array}]$

Step 15.16.3

Combine $\frac{- 7}{58 (x + 3 y)}$ and $y$ .

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3 x}{29 (x + 3 y)} + \frac{- 7 y}{58 (x + 3 y)} \end{array}]$

Step 15.17

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{58 (x + 3 y)} & - \frac{25}{58 (x + 3 y)} \\ - \frac{2 x}{29 (x + 3 y)} - \frac{5 y}{58 (x + 3 y)} & \frac{3 x}{29 (x + 3 y)} - \frac{7 y}{58 (x + 3 y)} \end{array}]$