Finite Math Examples

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

Step 1

Consider the corresponding sign chart.

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

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

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Step 2.1

Calculate the minor for element $a_{11}$ .

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Step 2.1.1

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 2.1.2

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

$a_{11} = e i - h f$

Step 2.2

Calculate the minor for element $a_{12}$ .

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Step 2.2.1

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 2.2.2

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

$a_{12} = d i - g f$

Step 2.3

Calculate the minor for element $a_{13}$ .

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Step 2.3.1

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 2.3.2

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

$a_{13} = d h - g e$

Step 2.4

Calculate the minor for element $a_{21}$ .

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Step 2.4.1

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

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

Step 2.4.2

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

$a_{21} = b i - h c$

Step 2.5

Calculate the minor for element $a_{22}$ .

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Step 2.5.1

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

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

Step 2.5.2

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

$a_{22} = a i - g c$

Step 2.6

Calculate the minor for element $a_{23}$ .

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Step 2.6.1

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} a & b \\ g & h \end{array} |$

Step 2.6.2

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

$a_{23} = a h - g b$

Step 2.7

Calculate the minor for element $a_{31}$ .

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Step 2.7.1

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

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

Step 2.7.2

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

$a_{31} = b f - e c$

Step 2.8

Calculate the minor for element $a_{32}$ .

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Step 2.8.1

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

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

Step 2.8.2

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

$a_{32} = a f - d c$

Step 2.9

Calculate the minor for element $a_{33}$ .

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Step 2.9.1

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Step 2.9.2

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

$a_{33} = a e - d b$

Step 2.10

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the $-$ positions on the sign chart.

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