Linear Algebra Examples

[\begin{array}{c} 5 & 2 \\ 4 & 3 \end{array}]

$[\begin{array}{c} 5 & 2 \\ 4 & 3 \end{array}]$

Step 1

Consider the corresponding sign chart.

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

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

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}

$a_{11}$ .

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Step 2.1.1

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} 3 \end{array} |

$| \begin{array}{c} 3 \end{array} |$

Step 2.1.2

The determinant of a

1 \times 1

$1 \times 1$ matrix is the element itself.

a_{11} = 3

$a_{11} = 3$

a_{11} = 3

$a_{11} = 3$

Step 2.2

Calculate the minor for element

a_{12}

$a_{12}$ .

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Step 2.2.1

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

| \begin{array}{c} 4 \end{array} |

$| \begin{array}{c} 4 \end{array} |$

Step 2.2.2

The determinant of a

1 \times 1

$1 \times 1$ matrix is the element itself.

a_{12} = 4

$a_{12} = 4$

a_{12} = 4

$a_{12} = 4$

Step 2.3

Calculate the minor for element

a_{21}

$a_{21}$ .

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Step 2.3.1

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

| \begin{array}{c} 2 \end{array} |

$| \begin{array}{c} 2 \end{array} |$

Step 2.3.2

The determinant of a

1 \times 1

$1 \times 1$ matrix is the element itself.

a_{21} = 2

$a_{21} = 2$

a_{21} = 2

$a_{21} = 2$

Step 2.4

Calculate the minor for element

a_{22}

$a_{22}$ .

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Step 2.4.1

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

| \begin{array}{c} 5 \end{array} |

$| \begin{array}{c} 5 \end{array} |$

Step 2.4.2

The determinant of a

1 \times 1

$1 \times 1$ matrix is the element itself.

a_{22} = 5

$a_{22} = 5$

a_{22} = 5

$a_{22} = 5$

Step 2.5

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} 3 & - 4 \\ - 2 & 5 \end{array}]

$[\begin{array}{c} 3 & - 4 \\ - 2 & 5 \end{array}]$

[\begin{array}{c} 3 & - 4 \\ - 2 & 5 \end{array}]

$[\begin{array}{c} 3 & - 4 \\ - 2 & 5 \end{array}]$