Linear Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} x & - \frac{1}{5} & - \frac{1}{5} - \frac{1}{5} & x & - \frac{1}{5} - \frac{1}{5} & - \frac{1}{5} & x \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} x & - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x & - \frac{1}{5} \\ - \frac{1}{5} & - \frac{1}{5} & x \end{array}]$

Step 1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 1.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

$| \begin{array}{c} x & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$

Step 1.4

Multiply element

$a_{11}$ by its cofactor.

$x | \begin{array}{c} x & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$

Step 1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

$| \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$

Step 1.6

Multiply element

$a_{12}$ by its cofactor.

$\frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$

Step 1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 1.8

Multiply element

$a_{13}$ by its cofactor.

$- \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 1.9

Add the terms together.

$x | \begin{array}{c} x & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2

Evaluate

$| \begin{array}{c} x & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$ .

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Step 2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

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

$x (x \cdot x - (- \frac{1}{5} (- \frac{1}{5}))) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply

$x$ by

$x$ .

$x (x^{2} - (- \frac{1}{5} (- \frac{1}{5}))) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2.2.2

Multiply

$- \frac{1}{5} (- \frac{1}{5})$ .

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Step 2.2.2.1

Multiply

$- 1$ by

$- 1$ .

$x (x^{2} - (1 (\frac{1}{5}) \frac{1}{5})) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2.2.2.2

Multiply

$\frac{1}{5}$ by

$1$ .

$x (x^{2} - (\frac{1}{5} \cdot \frac{1}{5})) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2.2.2.3

Multiply

$\frac{1}{5}$ by

$\frac{1}{5}$ .

$x (x^{2} - \frac{1}{5 \cdot 5}) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 2.2.2.4

Multiply

$5$ by

$5$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} | - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3

Evaluate

$| \begin{array}{c} - \frac{1}{5} & - \frac{1}{5} \\ - \frac{1}{5} & x \end{array} |$ .

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Step 3.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

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

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{1}{5} x - (- \frac{1}{5} (- \frac{1}{5}))) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine

$x$ and

$\frac{1}{5}$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - (- \frac{1}{5} (- \frac{1}{5}))) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3.2.2

Multiply

$- \frac{1}{5} (- \frac{1}{5})$ .

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Step 3.2.2.1

Multiply

$- 1$ by

$- 1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - (1 (\frac{1}{5}) \frac{1}{5})) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3.2.2.2

Multiply

$\frac{1}{5}$ by

$1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - (\frac{1}{5} \cdot \frac{1}{5})) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3.2.2.3

Multiply

$\frac{1}{5}$ by

$\frac{1}{5}$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{5 \cdot 5}) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 3.2.2.4

Multiply

$5$ by

$5$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} | \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$

Step 4

Evaluate

$| \begin{array}{c} - \frac{1}{5} & x \\ - \frac{1}{5} & - \frac{1}{5} \end{array} |$ .

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Step 4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

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

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (- \frac{1}{5} (- \frac{1}{5}) - (- \frac{1}{5} x))$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply

$- \frac{1}{5} (- \frac{1}{5})$ .

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Step 4.2.1.1

Multiply

$- 1$ by

$- 1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (1 (\frac{1}{5}) \frac{1}{5} - (- \frac{1}{5} x))$

Step 4.2.1.2

Multiply

$\frac{1}{5}$ by

$1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{5} \cdot \frac{1}{5} - (- \frac{1}{5} x))$

Step 4.2.1.3

Multiply

$\frac{1}{5}$ by

$\frac{1}{5}$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{5 \cdot 5} - (- \frac{1}{5} x))$

Step 4.2.1.4

Multiply

$5$ by

$5$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} - (- \frac{1}{5} x))$

Step 4.2.2

Combine

$x$ and

$\frac{1}{5}$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} - - \frac{x}{5})$

Step 4.2.3

Multiply

$- - \frac{x}{5}$ .

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Step 4.2.3.1

Multiply

$- 1$ by

$- 1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + 1 \frac{x}{5})$

Step 4.2.3.2

Multiply

$\frac{x}{5}$ by

$1$ .

$x (x^{2} - \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$x \cdot x^{2} + x (- \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.2

Multiply

$x$ by

$x^{2}$ by adding the exponents.

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Step 5.1.2.1

Multiply

$x$ by

$x^{2}$ .

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Step 5.1.2.1.1

Raise

$x$ to the power of

$1$ .

$x^{1} x^{2} + x (- \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.2.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 2} + x (- \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.2.2

Add

$1$ and

$2$ .

$x^{3} + x (- \frac{1}{25}) + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.3

Combine

$x$ and

$\frac{1}{25}$ .

$x^{3} - \frac{x}{25} + \frac{1}{5} (- \frac{x}{5} - \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.4

Apply the distributive property.

$x^{3} - \frac{x}{25} + \frac{1}{5} (- \frac{x}{5}) + \frac{1}{5} (- \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.5

Multiply

$\frac{1}{5} (- \frac{x}{5})$ .

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Step 5.1.5.1

Multiply

$\frac{1}{5}$ by

$\frac{x}{5}$ .

$x^{3} - \frac{x}{25} - \frac{x}{5 \cdot 5} + \frac{1}{5} (- \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.5.2

Multiply

$5$ by

$5$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} + \frac{1}{5} (- \frac{1}{25}) - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.6

Multiply

$\frac{1}{5} (- \frac{1}{25})$ .

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Step 5.1.6.1

Multiply

$\frac{1}{5}$ by

$\frac{1}{25}$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{5 \cdot 25} - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.6.2

Multiply

$5$ by

$25$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{5} (\frac{1}{25} + \frac{x}{5})$

Step 5.1.7

Apply the distributive property.

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{5} \cdot \frac{1}{25} - \frac{1}{5} \cdot \frac{x}{5}$

Step 5.1.8

Multiply

$- \frac{1}{5} \cdot \frac{1}{25}$ .

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Step 5.1.8.1

Multiply

$\frac{1}{25}$ by

$\frac{1}{5}$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{25 \cdot 5} - \frac{1}{5} \cdot \frac{x}{5}$

Step 5.1.8.2

Multiply

$25$ by

$5$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{125} - \frac{1}{5} \cdot \frac{x}{5}$

Step 5.1.9

Multiply

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

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Step 5.1.9.1

Multiply

$\frac{x}{5}$ by

$\frac{1}{5}$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{125} - \frac{x}{5 \cdot 5}$

Step 5.1.9.2

Multiply

$5$ by

$5$ .

$x^{3} - \frac{x}{25} - \frac{x}{25} - \frac{1}{125} - \frac{1}{125} - \frac{x}{25}$

Step 5.2

Combine the numerators over the common denominator.

$x^{3} + \frac{- x - x - x}{25} + \frac{- 1 - 1}{125}$

Step 5.3

Subtract

$x$ from

$- x$ .

$x^{3} + \frac{- 2 x - x}{25} + \frac{- 1 - 1}{125}$

Step 5.4

Subtract

$x$ from

$- 2 x$ .

$x^{3} + \frac{- 3 x}{25} + \frac{- 1 - 1}{125}$

Step 5.5

Subtract

$1$ from

$- 1$ .

$x^{3} + \frac{- 3 x}{25} + \frac{- 2}{125}$

Step 5.6

Simplify each term.

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Step 5.6.1

Move the negative in front of the fraction.

$x^{3} - \frac{3 x}{25} + \frac{- 2}{125}$

Step 5.6.2

Move the negative in front of the fraction.

$x^{3} - \frac{3 x}{25} - \frac{2}{125}$