Linear Algebra Examples

$[\begin{array}{c} a & - 2 b & 3 d \\ 4 a & b & - d \\ 2 a & - b & 3 d \end{array}]$

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.

$| \begin{array}{c} b & - d \\ - b & 3 d \end{array} |$

Step 1.4

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

$a | \begin{array}{c} b & - d \\ - b & 3 d \end{array} |$

Step 1.5

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

$| \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} |$

Step 1.6

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

$2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} |$

Step 1.7

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

$| \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 1.8

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

$3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 1.9

Add the terms together.

$a | \begin{array}{c} b & - d \\ - b & 3 d \end{array} | + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 2

Evaluate $| \begin{array}{c} b & - d \\ - b & 3 d \end{array} |$ .

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Step 2.1

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

$a (b (3 d) - (- b (- d))) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

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

Rewrite using the commutative property of multiplication.

$a (3 b d - (- b (- d))) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 2.2.1.2

Rewrite using the commutative property of multiplication.

$a (3 b d - (- 1 \cdot - 1 b d)) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 2.2.1.3

Multiply $- 1$ by $- 1$ .

$a (3 b d - (1 b d)) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 2.2.1.4

Multiply $b$ by $1$ .

$a (3 b d - b d) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 2.2.2

Subtract $b d$ from $3 b d$ .

$a (2 b d) + 2 b | \begin{array}{c} 4 a & - d \\ 2 a & 3 d \end{array} | + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 3

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

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Step 3.1

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

$a (2 b d) + 2 b (4 a (3 d) - 2 a (- d)) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

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

Rewrite using the commutative property of multiplication.

$a (2 b d) + 2 b (4 \cdot 3 a d - 2 a (- d)) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 3.2.1.2

Multiply $4$ by $3$ .

$a (2 b d) + 2 b (12 a d - 2 a (- d)) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

$a (2 b d) + 2 b (12 a d - 2 \cdot - 1 a d) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 3.2.1.4

Multiply $- 2$ by $- 1$ .

$a (2 b d) + 2 b (12 a d + 2 a d) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 3.2.2

Add $12 a d$ and $2 a d$ .

$a (2 b d) + 2 b (14 a d) + 3 d | \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 4 a & b \\ 2 a & - b \end{array} |$ .

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Step 4.1

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

$a (2 b d) + 2 b (14 a d) + 3 d (4 a (- b) - 2 a b)$

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

Rewrite using the commutative property of multiplication.

$a (2 b d) + 2 b (14 a d) + 3 d (4 \cdot - 1 a b - 2 a b)$

Step 4.2.1.2

Multiply $4$ by $- 1$ .

$a (2 b d) + 2 b (14 a d) + 3 d (- 4 a b - 2 a b)$

Step 4.2.2

Subtract $2 a b$ from $- 4 a b$ .

$a (2 b d) + 2 b (14 a d) + 3 d (- 6 a b)$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$2 a b d + 2 b (14 a d) + 3 d (- 6 a b)$

Step 5.1.2

Multiply $14$ by $2$ .

$2 a b d + 28 b (a d) + 3 d (- 6 a b)$

Step 5.1.3

Multiply $- 6$ by $3$ .

$2 a b d + 28 b a d - 18 d a b$

Step 5.2

Add $2 a b d$ and $28 b a d$ .

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Step 5.2.1

Move $b$ .

$2 a b d + 28 a b d - 18 d a b$

Step 5.2.2

Add $2 a b d$ and $28 a b d$ .

$30 a b d - 18 d a b$

Step 5.3

Subtract $18 d a b$ from $30 a b d$ .

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Step 5.3.1

Move $d$ .

$30 a b d - 18 a b d$

Step 5.3.2

Subtract $18 a b d$ from $30 a b d$ .

$12 a b d$