Linear Algebra Examples

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

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 column $1$ 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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

$- 4 | \begin{array}{c} 1 & 1 \\ 2 - x & 1 \end{array} |$

Step 1.9

Add the terms together.

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

Step 2

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

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

Step 3

Evaluate $| \begin{array}{c} 2 - x & 1 \\ 4 & 3 - x \end{array} |$ .

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Step 3.1

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

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

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

Expand $(2 - x) (3 - x)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Apply the distributive property.

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Multiply $2$ by $3$ .

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

Step 3.2.1.2.1.2

Multiply $- 1$ by $2$ .

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

Step 3.2.1.2.1.3

Multiply $3$ by $- 1$ .

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

Step 3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.5

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

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Step 3.2.1.2.1.5.1

Move $x$ .

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

Step 3.2.1.2.1.5.2

Multiply $x$ by $x$ .

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

Step 3.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.2.1.7

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

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

Step 3.2.1.2.2

Subtract $3 x$ from $- 2 x$ .

$(1 - x) (6 - 5 x + x^{2} - 4 \cdot 1) + 0 - 4 | \begin{array}{c} 1 & 1 \\ 2 - x & 1 \end{array} |$

Step 3.2.1.3

Multiply $- 4$ by $1$ .

$(1 - x) (6 - 5 x + x^{2} - 4) + 0 - 4 | \begin{array}{c} 1 & 1 \\ 2 - x & 1 \end{array} |$

Step 3.2.2

Subtract $4$ from $6$ .

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

Step 4

Evaluate $| \begin{array}{c} 1 & 1 \\ 2 - x & 1 \end{array} |$ .

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Step 4.1

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

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 \cdot 1 - (2 - x) \cdot 1)$

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 $1$ .

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 - (2 - x) \cdot 1)$

Step 4.2.1.2

Apply the distributive property.

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 + (- 1 \cdot 2 - - x) \cdot 1)$

Step 4.2.1.3

Multiply $- 1$ by $2$ .

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 + (- 2 - - x) \cdot 1)$

Step 4.2.1.4

Multiply $- - x$ .

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Step 4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 + (- 2 + 1 x) \cdot 1)$

Step 4.2.1.4.2

Multiply $x$ by $1$ .

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 + (- 2 + x) \cdot 1)$

Step 4.2.1.5

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

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (1 - 2 + x)$

Step 4.2.2

Subtract $2$ from $1$ .

$(1 - x) (- 5 x + x^{2} + 2) + 0 - 4 (- 1 + x)$

Step 5

Simplify the determinant.

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Step 5.1

Add $(1 - x) (- 5 x + x^{2} + 2)$ and $0$ .

$(1 - x) (- 5 x + x^{2} + 2) - 4 (- 1 + x)$

Step 5.2

Simplify each term.

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Step 5.2.1

Expand $(1 - x) (- 5 x + x^{2} + 2)$ by multiplying each term in the first expression by each term in the second expression.

$1 (- 5 x) + 1 x^{2} + 1 \cdot 2 - x (- 5 x) - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $- 5 x$ by $1$ .

$- 5 x + 1 x^{2} + 1 \cdot 2 - x (- 5 x) - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.2

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

$- 5 x + x^{2} + 1 \cdot 2 - x (- 5 x) - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.3

Multiply $2$ by $1$ .

$- 5 x + x^{2} + 2 - x (- 5 x) - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.4

Rewrite using the commutative property of multiplication.

$- 5 x + x^{2} + 2 - 1 \cdot - 5 x \cdot x - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.5

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

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Step 5.2.2.5.1

Move $x$ .

$- 5 x + x^{2} + 2 - 1 \cdot - 5 (x \cdot x) - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.5.2

Multiply $x$ by $x$ .

$- 5 x + x^{2} + 2 - 1 \cdot - 5 x^{2} - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.6

Multiply $- 1$ by $- 5$ .

$- 5 x + x^{2} + 2 + 5 x^{2} - x \cdot x^{2} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.7

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

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Step 5.2.2.7.1

Move $x^{2}$ .

$- 5 x + x^{2} + 2 + 5 x^{2} - (x^{2} x) - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.7.2

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

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Step 5.2.2.7.2.1

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

$- 5 x + x^{2} + 2 + 5 x^{2} - (x^{2} x^{1}) - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.7.2.2

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

$- 5 x + x^{2} + 2 + 5 x^{2} - x^{2 + 1} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.7.3

Add $2$ and $1$ .

$- 5 x + x^{2} + 2 + 5 x^{2} - x^{3} - x \cdot 2 - 4 (- 1 + x)$

Step 5.2.2.8

Multiply $2$ by $- 1$ .

$- 5 x + x^{2} + 2 + 5 x^{2} - x^{3} - 2 x - 4 (- 1 + x)$

Step 5.2.3

Subtract $2 x$ from $- 5 x$ .

$x^{2} + 2 + 5 x^{2} - x^{3} - 7 x - 4 (- 1 + x)$

Step 5.2.4

Add $x^{2}$ and $5 x^{2}$ .

$6 x^{2} + 2 - x^{3} - 7 x - 4 (- 1 + x)$

Step 5.2.5

Apply the distributive property.

$6 x^{2} + 2 - x^{3} - 7 x - 4 \cdot - 1 - 4 x$

Step 5.2.6

Multiply $- 4$ by $- 1$ .

$6 x^{2} + 2 - x^{3} - 7 x + 4 - 4 x$

Step 5.3

Add $2$ and $4$ .

$6 x^{2} - x^{3} - 7 x + 6 - 4 x$

Step 5.4

Subtract $4 x$ from $- 7 x$ .

$6 x^{2} - x^{3} - 11 x + 6$