Linear Algebra Examples

A = ⎡ ⎢ ⎣ \begin{matrix} x & 3 & x^{2} - 3 & 5 x & 0 4 & x^{3} & 1 \end{matrix} ⎤ ⎥ ⎦

$A = [\begin{array}{c} x & 3 & x^{2} \\ - 3 & 5 x & 0 \\ 4 & x^{3} & 1 \end{array}]$

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.

$| \begin{array}{c} 5 x & 0 \\ x^{3} & 1 \end{array} |$

Step 1.4

Multiply element

$a_{11}$ by its cofactor.

$x | \begin{array}{c} 5 x & 0 \\ x^{3} & 1 \end{array} |$

Step 1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

$| \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} |$

Step 1.6

Multiply element

$a_{12}$ by its cofactor.

$- 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} |$

Step 1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 1.8

Multiply element

$a_{13}$ by its cofactor.

$x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 1.9

Add the terms together.

$x | \begin{array}{c} 5 x & 0 \\ x^{3} & 1 \end{array} | - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 2

Evaluate

$| \begin{array}{c} 5 x & 0 \\ x^{3} & 1 \end{array} |$ .

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Step 2.1

The determinant of a

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

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

$x (5 x \cdot 1 - x^{3} \cdot 0) - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

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

Multiply

$5$ by

$1$ .

$x (5 x - x^{3} \cdot 0) - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 2.2.1.2

Multiply

$- x^{3} \cdot 0$ .

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Step 2.2.1.2.1

Multiply

$0$ by

$- 1$ .

$x (5 x + 0 x^{3}) - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 2.2.1.2.2

Multiply

$0$ by

$x^{3}$ .

$x (5 x + 0) - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 2.2.2

Add

$5 x$ and

$0$ .

$x (5 x) - 3 | \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} | + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 3

Evaluate

$| \begin{array}{c} - 3 & 0 \\ 4 & 1 \end{array} |$ .

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Step 3.1

The determinant of a

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

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

$x (5 x) - 3 (- 3 \cdot 1 - 4 \cdot 0) + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply

$- 3$ by

$1$ .

$x (5 x) - 3 (- 3 - 4 \cdot 0) + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 3.2.1.2

Multiply

$- 4$ by

$0$ .

$x (5 x) - 3 (- 3 + 0) + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 3.2.2

Add

$- 3$ and

$0$ .

$x (5 x) - 3 \cdot - 3 + x^{2} | \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$

Step 4

Evaluate

$| \begin{array}{c} - 3 & 5 x \\ 4 & x^{3} \end{array} |$ .

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Step 4.1

The determinant of a

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

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

$x (5 x) - 3 \cdot - 3 + x^{2} (- 3 x^{3} - 4 (5 x))$

Step 4.2

Multiply

$5$ by

$- 4$ .

$x (5 x) - 3 \cdot - 3 + x^{2} (- 3 x^{3} - 20 x)$

Step 5

Simplify each term.

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Step 5.1

Rewrite using the commutative property of multiplication.

$5 x \cdot x - 3 \cdot - 3 + x^{2} (- 3 x^{3} - 20 x)$

Step 5.2

Multiply

$x$ by

$x$ by adding the exponents.

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Step 5.2.1

Move

$x$ .

$5 (x \cdot x) - 3 \cdot - 3 + x^{2} (- 3 x^{3} - 20 x)$

Step 5.2.2

Multiply

$x$ by

$x$ .

$5 x^{2} - 3 \cdot - 3 + x^{2} (- 3 x^{3} - 20 x)$

Step 5.3

Multiply

$- 3$ by

$- 3$ .

$5 x^{2} + 9 + x^{2} (- 3 x^{3} - 20 x)$

Step 5.4

Apply the distributive property.

$5 x^{2} + 9 + x^{2} (- 3 x^{3}) + x^{2} (- 20 x)$

Step 5.5

Rewrite using the commutative property of multiplication.

$5 x^{2} + 9 - 3 x^{2} x^{3} + x^{2} (- 20 x)$

Step 5.6

Rewrite using the commutative property of multiplication.

$5 x^{2} + 9 - 3 x^{2} x^{3} - 20 x^{2} x$

Step 5.7

Simplify each term.

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Step 5.7.1

Multiply

$x^{2}$ by

$x^{3}$ by adding the exponents.

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Step 5.7.1.1

Move

$x^{3}$ .

$5 x^{2} + 9 - 3 (x^{3} x^{2}) - 20 x^{2} x$

Step 5.7.1.2

Use the power rule

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

$5 x^{2} + 9 - 3 x^{3 + 2} - 20 x^{2} x$

Step 5.7.1.3

Add

$3$ and

$2$ .

$5 x^{2} + 9 - 3 x^{5} - 20 x^{2} x$

Step 5.7.2

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 5.7.2.1

Move

$x$ .

$5 x^{2} + 9 - 3 x^{5} - 20 (x \cdot x^{2})$

Step 5.7.2.2

Multiply

$x$ by

$x^{2}$ .

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Step 5.7.2.2.1

Raise

$x$ to the power of

$1$ .

$5 x^{2} + 9 - 3 x^{5} - 20 (x^{1} x^{2})$

Step 5.7.2.2.2

Use the power rule

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

$5 x^{2} + 9 - 3 x^{5} - 20 x^{1 + 2}$

Step 5.7.2.3

Add

$1$ and

$2$ .

$5 x^{2} + 9 - 3 x^{5} - 20 x^{3}$