Finite Math Examples

[\begin{matrix} cos (45) & sin (60) sin (60) & cos (- 45) \end{matrix}]

$[\begin{array}{c} \cos (45) & \sin (60) \\ \sin (60) & \cos (- 45) \end{array}]$

Step 1

Consider the corresponding sign chart.

[\begin{matrix} + & - - & + \end{matrix}]

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

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{matrix} cos (- 45) \end{matrix} |

$| \begin{array}{c} \cos (- 45) \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a

1 \times 1

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

a_{11} = cos (- 45)

$a_{11} = \cos (- 45)$

Step 2.1.2.2

Simplify the determinant.

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Step 2.1.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

a_{11} = cos (45)

$a_{11} = \cos (45)$

Step 2.1.2.2.2

The exact value of

cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

a_{11} = \frac{\sqrt{2}}{2}

$a_{11} = \frac{\sqrt{2}}{2}$

a_{11} = \frac{\sqrt{2}}{2}

$a_{11} = \frac{\sqrt{2}}{2}$

a_{11} = \frac{\sqrt{2}}{2}

$a_{11} = \frac{\sqrt{2}}{2}$

a_{11} = \frac{\sqrt{2}}{2}

$a_{11} = \frac{\sqrt{2}}{2}$

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

$1$ and column

$2$ deleted.

$| \begin{array}{c} \sin (60) \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a

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

$a_{12} = \sin (60)$

Step 2.2.2.2

The exact value of

$\sin (60)$ is

$\frac{\sqrt{3}}{2}$ .

$a_{12} = \frac{\sqrt{3}}{2}$

Step 2.3

Calculate the minor for element

$a_{21}$ .

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Step 2.3.1

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} \sin (60) \end{array} |$

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

The determinant of a

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

$a_{21} = \sin (60)$

Step 2.3.2.2

The exact value of

$\sin (60)$ is

$\frac{\sqrt{3}}{2}$ .

$a_{21} = \frac{\sqrt{3}}{2}$

Step 2.4

Calculate the minor for element

$a_{22}$ .

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Step 2.4.1

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} \cos (45) \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

The determinant of a

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

$a_{22} = \cos (45)$

Step 2.4.2.2

The exact value of

$\cos (45)$ is

$\frac{\sqrt{2}}{2}$ .

$a_{22} = \frac{\sqrt{2}}{2}$

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.

$[\begin{array}{c} \frac{\sqrt{2}}{2} & - \frac{\sqrt{3}}{2} \\ - \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} \frac{\sqrt{2}}{2} & - \frac{\sqrt{3}}{2} \\ - \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} \end{array}]$