Linear Algebra Examples

$[\begin{array}{c} a & b & c \\ d & e & f \\ 4 g & 4 h & 4 i \end{array}]$

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.

$| \begin{array}{c} e & f \\ 4 h & 4 i \end{array} |$

Step 1.4

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

$a | \begin{array}{c} e & f \\ 4 h & 4 i \end{array} |$

Step 1.5

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

$| \begin{array}{c} d & f \\ 4 g & 4 i \end{array} |$

Step 1.6

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

$- b | \begin{array}{c} d & f \\ 4 g & 4 i \end{array} |$

Step 1.7

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

$| \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 1.8

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

$c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 1.9

Add the terms together.

$a | \begin{array}{c} e & f \\ 4 h & 4 i \end{array} | - b | \begin{array}{c} d & f \\ 4 g & 4 i \end{array} | + c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 2

Evaluate $| \begin{array}{c} e & f \\ 4 h & 4 i \end{array} |$ .

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Step 2.1

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

$a (e (4 i) - 4 h f) - b | \begin{array}{c} d & f \\ 4 g & 4 i \end{array} | + c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 2.2

Move $4$ to the left of $e$ .

$a (4 e i - 4 h f) - b | \begin{array}{c} d & f \\ 4 g & 4 i \end{array} | + c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 3

Evaluate $| \begin{array}{c} d & f \\ 4 g & 4 i \end{array} |$ .

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Step 3.1

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

$a (4 e i - 4 h f) - b (d (4 i) - 4 g f) + c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 3.2

Move $4$ to the left of $d$ .

$a (4 e i - 4 h f) - b (4 d i - 4 g f) + c | \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$

Step 4

Evaluate $| \begin{array}{c} d & e \\ 4 g & 4 h \end{array} |$ .

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Step 4.1

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

$a (4 e i - 4 h f) - b (4 d i - 4 g f) + c (d (4 h) - 4 g e)$

Step 4.2

Rewrite using the commutative property of multiplication.

$a (4 e i - 4 h f) - b (4 d i - 4 g f) + c (4 d h - 4 g e)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a (4 e i) + a (- 4 h f) - b (4 d i - 4 g f) + c (4 d h - 4 g e)$

Step 5.2

Rewrite using the commutative property of multiplication.

$a \cdot 4 e i - 4 a h f - b (4 d i - 4 g f) + c (4 d h - 4 g e)$

Step 5.3

Move $4$ to the left of $a$ .

$4 a e i - 4 a h f - b (4 d i - 4 g f) + c (4 d h - 4 g e)$

Step 5.4

Apply the distributive property.

$4 a e i - 4 a h f - b (4 d i) - b (- 4 g f) + c (4 d h - 4 g e)$

Step 5.5

Multiply $4$ by $- 1$ .

$4 a e i - 4 a h f - 4 b (d i) - b (- 4 g f) + c (4 d h - 4 g e)$

Step 5.6

Multiply $- 4$ by $- 1$ .

$4 a e i - 4 a h f - 4 b (d i) + 4 b (g f) + c (4 d h - 4 g e)$

Step 5.7

Remove parentheses.

$4 a e i - 4 a h f - 4 b d i + 4 b g f + c (4 d h - 4 g e)$

Step 5.8

Apply the distributive property.

$4 a e i - 4 a h f - 4 b d i + 4 b g f + c (4 d h) + c (- 4 g e)$

Step 5.9

Rewrite using the commutative property of multiplication.

$4 a e i - 4 a h f - 4 b d i + 4 b g f + 4 c d h + c (- 4 g e)$

Step 5.10

Rewrite using the commutative property of multiplication.

$4 a e i - 4 a h f - 4 b d i + 4 b g f + 4 c d h - 4 e c g$