Finite Math Examples

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add the terms together.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- b (- g f)$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

$a e i - a h f - b d i + 1 b (g f) + c (d h - g e)$

Step 5.4.2

Multiply $b$ by $1$ .

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

Step 5.5

Apply the distributive property.

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

Step 5.6

Rewrite using the commutative property of multiplication.

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