Linear Algebra Examples

$[\begin{array}{c} x & 1 & 0 & 0 \\ 1 & x & 1 & 1 \\ 0 & 1 & x & 1 \\ 0 & 1 & 1 & x \end{array}]$

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

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

Step 1.10

Multiply element $a_{14}$ by its cofactor.

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

Step 1.11

Add the terms together.

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

Step 2

Multiply $0$ by $| \begin{array}{c} 1 & x & 1 \\ 0 & 1 & 1 \\ 0 & 1 & x \end{array} |$ .

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

Step 3

Multiply $0$ by $| \begin{array}{c} 1 & x & 1 \\ 0 & 1 & x \\ 0 & 1 & 1 \end{array} |$ .

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

Step 4

Evaluate $| \begin{array}{c} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{array} |$ .

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Step 4.1

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 4.1.1

Consider the corresponding sign chart.

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

Step 4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

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

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

Step 4.1.8

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

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

Step 4.1.9

Add the terms together.

$x (x | \begin{array}{c} x & 1 \\ 1 & x \end{array} | - 1 | \begin{array}{c} 1 & 1 \\ 1 & x \end{array} | + 1 | \begin{array}{c} 1 & x \\ 1 & 1 \end{array} |) - 1 | \begin{array}{c} 1 & 1 & 1 \\ 0 & x & 1 \\ 0 & 1 & x \end{array} | + 0 + 0$

Step 4.2

Evaluate $| \begin{array}{c} x & 1 \\ 1 & x \end{array} |$ .

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Step 4.2.1

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

$x (x (x \cdot x - 1 \cdot 1) - 1 | \begin{array}{c} 1 & 1 \\ 1 & x \end{array} | + 1 | \begin{array}{c} 1 & x \\ 1 & 1 \end{array} |) - 1 | \begin{array}{c} 1 & 1 & 1 \\ 0 & x & 1 \\ 0 & 1 & x \end{array} | + 0 + 0$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply $x$ by $x$ .

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

Step 4.2.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3

Evaluate $| \begin{array}{c} 1 & 1 \\ 1 & x \end{array} |$ .

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Step 4.3.1

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

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply $x$ by $1$ .

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

Step 4.3.2.2

Multiply $- 1$ by $1$ .

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

Step 4.4

Evaluate $| \begin{array}{c} 1 & x \\ 1 & 1 \end{array} |$ .

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Step 4.4.1

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

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

Step 4.4.2

Multiply $1$ by $1$ .

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

Step 4.5

Simplify the determinant.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Apply the distributive property.

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

Step 4.5.1.2

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

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Step 4.5.1.2.1

Multiply $x$ by $x^{2}$ .

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Step 4.5.1.2.1.1

Raise $x$ to the power of $1$ .

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

Step 4.5.1.2.1.2

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

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

Step 4.5.1.2.2

Add $1$ and $2$ .

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

Step 4.5.1.3

Move $- 1$ to the left of $x$ .

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

Step 4.5.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 4.5.1.5

Apply the distributive property.

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

Step 4.5.1.6

Rewrite $- 1 x$ as $- x$ .

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

Step 4.5.1.7

Multiply $- 1$ by $- 1$ .

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

Step 4.5.1.8

Multiply $1 - x$ by $1$ .

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

Step 4.5.2

Subtract $x$ from $- x$ .

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

Step 4.5.3

Subtract $x$ from $- 2 x$ .

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

Step 4.5.4

Add $1$ and $1$ .

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

Step 5

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

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Step 5.1

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 column $1$ by its cofactor and add.

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Step 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

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

Step 5.1.4

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

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

Step 5.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 5.1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Step 5.1.8

Multiply element $a_{31}$ by its cofactor.

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

Step 5.1.9

Add the terms together.

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

Step 5.2

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ 1 & x \end{array} |$ .

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

Step 5.3

Multiply $0$ by $| \begin{array}{c} 1 & 1 \\ x & 1 \end{array} |$ .

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

Step 5.4

Evaluate $| \begin{array}{c} x & 1 \\ 1 & x \end{array} |$ .

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Step 5.4.1

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

$x (x^{3} - 3 x + 2) - 1 (1 (x \cdot x - 1 \cdot 1) + 0 + 0) + 0 + 0$

Step 5.4.2

Simplify each term.

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Step 5.4.2.1

Multiply $x$ by $x$ .

$x (x^{3} - 3 x + 2) - 1 (1 (x^{2} - 1 \cdot 1) + 0 + 0) + 0 + 0$

Step 5.4.2.2

Multiply $- 1$ by $1$ .

$x (x^{3} - 3 x + 2) - 1 (1 (x^{2} - 1) + 0 + 0) + 0 + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Combine the opposite terms in $1 (x^{2} - 1) + 0 + 0$ .

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Step 5.5.1.1

Add $1 (x^{2} - 1)$ and $0$ .

$x (x^{3} - 3 x + 2) - 1 (1 (x^{2} - 1) + 0) + 0 + 0$

Step 5.5.1.2

Add $1 (x^{2} - 1)$ and $0$ .

$x (x^{3} - 3 x + 2) - 1 (1 (x^{2} - 1)) + 0 + 0$

Step 5.5.2

Multiply $x^{2} - 1$ by $1$ .

$x (x^{3} - 3 x + 2) - 1 (x^{2} - 1) + 0 + 0$

Step 6

Simplify the determinant.

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Step 6.1

Combine the opposite terms in $x (x^{3} - 3 x + 2) - 1 (x^{2} - 1) + 0 + 0$ .

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Step 6.1.1

Add $x (x^{3} - 3 x + 2) - 1 (x^{2} - 1)$ and $0$ .

$x (x^{3} - 3 x + 2) - 1 (x^{2} - 1) + 0$

Step 6.1.2

Add $x (x^{3} - 3 x + 2) - 1 (x^{2} - 1)$ and $0$ .

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

Step 6.2

Simplify each term.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Simplify.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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Step 6.2.2.1.1.1

Raise $x$ to the power of $1$ .

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

Step 6.2.2.1.1.2

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

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

Step 6.2.2.1.2

Add $1$ and $3$ .

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

Step 6.2.2.2

Rewrite using the commutative property of multiplication.

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

Step 6.2.2.3

Move $2$ to the left of $x$ .

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

Step 6.2.3

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

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Step 6.2.3.1

Move $x$ .

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

Step 6.2.3.2

Multiply $x$ by $x$ .

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

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

Step 6.2.4

Apply the distributive property.

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

Step 6.2.5

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 6.2.6

Multiply $- 1$ by $- 1$ .

$x^{4} - 3 x^{2} + 2 x - x^{2} + 1$

Step 6.3

Subtract $x^{2}$ from $- 3 x^{2}$ .

$x^{4} - 4 x^{2} + 2 x + 1$