Linear Algebra Examples

$[\begin{array}{c} x & 4 & x^{2} \\ 6 & - x & 0 \\ - 1 & x^{3} & 1 \end{array}]$

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.

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

Step 1.4

Multiply element $a_{11}$ by its cofactor.

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

Step 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} |$

Step 1.6

Multiply element $a_{12}$ by its cofactor.

$- 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} |$

Step 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 1.8

Multiply element $a_{13}$ by its cofactor.

$x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 1.9

Add the terms together.

$x | \begin{array}{c} - x & 0 \\ x^{3} & 1 \end{array} | - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 2

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

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Step 2.1

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

$x (- x \cdot 1 - x^{3} \cdot 0) - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

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 $- 1$ by $1$ .

$x (- x - x^{3} \cdot 0) - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Multiply $0$ by $- 1$ .

$x (- x + 0 x^{3}) - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 2.2.1.2.2

Multiply $0$ by $x^{3}$ .

$x (- x + 0) - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 2.2.2

Add $- x$ and $0$ .

$x (- x) - 4 | \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} | + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 6 & 0 \\ - 1 & 1 \end{array} |$ .

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Step 3.1

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

$x (- x) - 4 (6 \cdot 1 - - 0) + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

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 $6$ by $1$ .

$x (- x) - 4 (6 - - 0) + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 3.2.1.2

Multiply $- - 0$ .

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Step 3.2.1.2.1

Multiply $- 1$ by $0$ .

$x (- x) - 4 (6 - 0) + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 3.2.1.2.2

Multiply $- 1$ by $0$ .

$x (- x) - 4 (6 + 0) + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 3.2.2

Add $6$ and $0$ .

$x (- x) - 4 \cdot 6 + x^{2} | \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 6 & - x \\ - 1 & x^{3} \end{array} |$ .

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Step 4.1

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

$x (- x) - 4 \cdot 6 + x^{2} (6 x^{3} - - - x)$

Step 4.2

Multiply $- - x$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$x (- x) - 4 \cdot 6 + x^{2} (6 x^{3} - (1 x))$

Step 4.2.2

Multiply $x$ by $1$ .

$x (- x) - 4 \cdot 6 + x^{2} (6 x^{3} - x)$

Step 5

Simplify each term.

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Step 5.1

Rewrite using the commutative property of multiplication.

$- x \cdot x - 4 \cdot 6 + x^{2} (6 x^{3} - x)$

Step 5.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1

Move $x$ .

$- (x \cdot x) - 4 \cdot 6 + x^{2} (6 x^{3} - x)$

Step 5.2.2

Multiply $x$ by $x$ .

$- x^{2} - 4 \cdot 6 + x^{2} (6 x^{3} - x)$

Step 5.3

Multiply $- 4$ by $6$ .

$- x^{2} - 24 + x^{2} (6 x^{3} - x)$

Step 5.4

Apply the distributive property.

$- x^{2} - 24 + x^{2} (6 x^{3}) + x^{2} (- x)$

Step 5.5

Rewrite using the commutative property of multiplication.

$- x^{2} - 24 + 6 x^{2} x^{3} + x^{2} (- x)$

Step 5.6

Rewrite using the commutative property of multiplication.

$- x^{2} - 24 + 6 x^{2} x^{3} - x^{2} x$

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 $x^{3}$ .

$- x^{2} - 24 + 6 (x^{3} x^{2}) - x^{2} x$

Step 5.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2} - 24 + 6 x^{3 + 2} - x^{2} x$

Step 5.7.1.3

Add $3$ and $2$ .

$- x^{2} - 24 + 6 x^{5} - x^{2} x$

Step 5.7.2

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

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Step 5.7.2.1

Move $x$ .

$- x^{2} - 24 + 6 x^{5} - (x \cdot x^{2})$

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 $1$ .

$- x^{2} - 24 + 6 x^{5} - (x^{1} x^{2})$

Step 5.7.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2} - 24 + 6 x^{5} - x^{1 + 2}$

Step 5.7.2.3

Add $1$ and $2$ .

$- x^{2} - 24 + 6 x^{5} - x^{3}$