Linear Algebra Examples

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin (x) \sin (x) & y \cos (x) \cos (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 1

Multiply $- y \sin (x) \sin (x)$ .

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Step 1.1

Raise $\sin (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y (\sin^{1} (x) \sin (x)) & y \cos (x) \cos (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 1.2

Raise $\sin (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y (\sin^{1} (x) \sin^{1} (x)) & y \cos (x) \cos (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} \cos (x) \sin (x) & - y {\sin (x)}^{1 + 1} & y \cos (x) \cos (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 1.4

Add $1$ and $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos (x) \cos (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 2

Multiply $y \cos (x) \cos (x)$ .

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Step 2.1

Raise $\cos (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y (\cos^{1} (x) \cos (x)) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 2.2

Raise $\cos (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y (\cos^{1} (x) \cos^{1} (x)) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y {\cos (x)}^{1 + 1} \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 2.4

Add $1$ and $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos^{2} (x) \\ \sin (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 3

Multiply $\sin (x) \sin (x)$ .

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Step 3.1

Raise $\sin (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos^{2} (x) \\ \sin^{1} (x) \sin (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 3.2

Raise $\sin (x)$ to the power of $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos^{2} (x) \\ \sin^{1} (x) \sin^{1} (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos^{2} (x) \\ {\sin (x)}^{1 + 1} & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$

Step 3.4

Add $1$ and $1$ .

$[\begin{array}{c} \cos (x) \sin (x) & - y \sin^{2} (x) & y \cos^{2} (x) \\ \sin^{2} (x) & y \cos (x) \sin (x) & y \sin (x) \cos (x) \\ \cos (x) & 0 & - y \sin (x) \end{array}]$