Linear Algebra Examples

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add the terms together.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $x$ by $x$ .

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $- 1$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $1$ .

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

Step 3.2.2

Multiply $- 1$ by $1$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $1$ by $1$ .

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

Step 4.2.1.2

Apply the distributive property.

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

Step 4.2.1.3

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2

Add $1$ and $1$ .

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

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Expand $(x + 1) (x^{2} - x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.1.2

Simplify each term.

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Step 5.1.2.1

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

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Step 5.1.2.1.1

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

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Step 5.1.2.1.1.1

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

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

Step 5.1.2.1.1.2

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

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

Step 5.1.2.1.2

Add $1$ and $2$ .

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

Step 5.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.2.3

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

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Step 5.1.2.3.1

Move $x$ .

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

Step 5.1.2.3.2

Multiply $x$ by $x$ .

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

Step 5.1.2.4

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

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

Step 5.1.2.5

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

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

Step 5.1.2.6

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

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

Step 5.1.2.7

Multiply $- x$ by $1$ .

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

Step 5.1.2.8

Multiply $- 1$ by $1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

Add $- x^{2}$ and $x^{2}$ .

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

Step 5.1.3.2

Add $x^{3} - x$ and $0$ .

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

Step 5.1.4

Subtract $x$ from $- x$ .

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

Step 5.1.5

Apply the distributive property.

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

Step 5.1.6

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

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

Step 5.1.7

Multiply $- 1$ by $- 1$ .

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

Step 5.1.8

Multiply $- x + 2$ by $1$ .

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

Step 5.2

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

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Step 5.2.1

Add $- 1$ and $1$ .

$x^{3} - 2 x - x + 0 - x + 2$

Step 5.2.2

Add $x^{3} - 2 x - x$ and $0$ .

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

Step 5.3

Subtract $x$ from $- 2 x$ .

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

Step 5.4

Subtract $x$ from $- 3 x$ .

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