Linear Algebra Examples

[\begin{matrix} cos (1) & sin (1) - sin (1) & cos (1) \end{matrix}]

$[\begin{array}{c} \cos (1) & \sin (1) \\ - \sin (1) & \cos (1) \end{array}]$

Step 1

Evaluate

cos (1)

$\cos (1)$ .

[\begin{matrix} 0.99984769 & sin (1) - sin (1) & cos (1) \end{matrix}]

$[\begin{array}{c} 0.99984769 & \sin (1) \\ - \sin (1) & \cos (1) \end{array}]$

Step 2

Evaluate

sin (1)

$\sin (1)$ .

[\begin{matrix} 0.99984769 & 0.0174524 - sin (1) & cos (1) \end{matrix}]

$[\begin{array}{c} 0.99984769 & 0.0174524 \\ - \sin (1) & \cos (1) \end{array}]$

Step 3

Evaluate

sin (1)

$\sin (1)$ .

[\begin{matrix} 0.99984769 & 0.0174524 - 1 \cdot 0.0174524 & cos (1) \end{matrix}]

$[\begin{array}{c} 0.99984769 & 0.0174524 \\ - 1 \cdot 0.0174524 & \cos (1) \end{array}]$

Step 4

Multiply

- 1

$- 1$ by

0.0174524

$0.0174524$ .

[\begin{matrix} 0.99984769 & 0.0174524 - 0.0174524 & cos (1) \end{matrix}]

$[\begin{array}{c} 0.99984769 & 0.0174524 \\ - 0.0174524 & \cos (1) \end{array}]$

Step 5

Evaluate

cos (1)

$\cos (1)$ .

[\begin{matrix} 0.99984769 & 0.0174524 - 0.0174524 & 0.99984769 \end{matrix}]

$[\begin{array}{c} 0.99984769 & 0.0174524 \\ - 0.0174524 & 0.99984769 \end{array}]$

Step 6

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{0.99984769}

$\frac{1}{0.99984769}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Step 6.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{0.99984769}

$\frac{1}{0.99984769}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{matrix} \frac{0.99984769}{0.99984769} & \frac{0.0174524}{0.99984769} - 0.0174524 & 0.99984769 \end{matrix}]

$[\begin{array}{c} \frac{0.99984769}{0.99984769} & \frac{0.0174524}{0.99984769} \\ - 0.0174524 & 0.99984769 \end{array}]$

Step 6.2

Simplify

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 0.01745506 - 0.0174524 & 0.99984769 \end{matrix}]

$[\begin{array}{c} 1 & 0.01745506 \\ - 0.0174524 & 0.99984769 \end{array}]$

[\begin{matrix} 1 & 0.01745506 - 0.0174524 & 0.99984769 \end{matrix}]

$[\begin{array}{c} 1 & 0.01745506 \\ - 0.0174524 & 0.99984769 \end{array}]$

Step 7

Perform the row operation

R_{2} = R_{2} + 0.0174524 R_{1}

$R_{2} = R_{2} + 0.0174524 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 7.1

Perform the row operation

R_{2} = R_{2} + 0.0174524 R_{1}

$R_{2} = R_{2} + 0.0174524 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{matrix} 1 & 0.01745506 - 0.0174524 + 0.0174524 \cdot 1 & 0.99984769 + 0.0174524 \cdot 0.01745506 \end{matrix}]

$[\begin{array}{c} 1 & 0.01745506 \\ - 0.0174524 + 0.0174524 \cdot 1 & 0.99984769 + 0.0174524 \cdot 0.01745506 \end{array}]$

Step 7.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & 0.01745506 0 & 1.00015232 \end{matrix}]

$[\begin{array}{c} 1 & 0.01745506 \\ 0 & 1.00015232 \end{array}]$

[\begin{matrix} 1 & 0.01745506 0 & 1.00015232 \end{matrix}]

$[\begin{array}{c} 1 & 0.01745506 \\ 0 & 1.00015232 \end{array}]$

Step 8

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{1.00015232}

$\frac{1}{1.00015232}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 8.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{1.00015232}

$\frac{1}{1.00015232}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & 0.01745506 \\ \frac{0}{1.00015232} & \frac{1.00015232}{1.00015232} \end{array}]$

Step 8.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & 0.01745506 \\ 0 & 1 \end{array}]$

Step 9

Perform the row operation

$R_{1} = R_{1} - 0.01745506 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 9.1

Perform the row operation

$R_{1} = R_{1} - 0.01745506 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - 0.01745506 \cdot 0 & 0.01745506 - 0.01745506 \cdot 1 \\ 0 & 1 \end{array}]$

Step 9.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$