Linear Algebra Examples

$[\begin{array}{c} \sin (t h e t a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 1

Multiply $t$ by $t$ by adding the exponents.

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Step 1.1

Move $t$ .

$[\begin{array}{c} \sin (t \cdot t h e a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 1.2

Multiply $t$ by $t$ .

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 2

Multiply $t$ by $t$ by adding the exponents.

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Step 2.1

Move $t$ .

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t \cdot t h e a) \end{array}]$

Step 2.2

Multiply $t$ by $t$ .

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t^{2} h e a) \end{array}]$

Step 3

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

$\sin (t^{2} h e a) \sin (t^{2} h e a) - - - 1$

Step 4

Simplify the determinant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\sin (t^{2} h e a) \sin (t^{2} h e a)$ .

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Step 4.1.1.1

Raise $\sin (t^{2} h e a)$ to the power of $1$ .

$\sin^{1} (t^{2} h e a) \sin (t^{2} h e a) - - - 1$

Step 4.1.1.2

Raise $\sin (t^{2} h e a)$ to the power of $1$ .

$\sin^{1} (t^{2} h e a) \sin^{1} (t^{2} h e a) - - - 1$

Step 4.1.1.3

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

${\sin (t^{2} h e a)}^{1 + 1} - - - 1$

Step 4.1.1.4

Add $1$ and $1$ .

$\sin^{2} (t^{2} h e a) - - - 1$

Step 4.1.2

Multiply $- - - 1$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\sin^{2} (t^{2} h e a) - 1 \cdot 1$

Step 4.1.2.2

Multiply $- 1$ by $1$ .

$\sin^{2} (t^{2} h e a) - 1$

Step 4.2

Reorder $\sin^{2} (t^{2} h e a)$ and $- 1$ .

$- 1 + \sin^{2} (t^{2} h e a)$

Step 4.3

Rewrite $- 1$ as $- 1 (1)$ .

$- 1 (1) + \sin^{2} (t^{2} h e a)$

Step 4.4

Factor $- 1$ out of $\sin^{2} (t^{2} h e a)$ .

$- 1 (1) - 1 (- \sin^{2} (t^{2} h e a))$

Step 4.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (t^{2} h e a))$ .

$- 1 (1 - \sin^{2} (t^{2} h e a))$

Step 4.6

Rewrite $- 1 (1 - \sin^{2} (t^{2} h e a))$ as $- (1 - \sin^{2} (t^{2} h e a))$ .

$- (1 - \sin^{2} (t^{2} h e a))$

Step 4.7

Apply pythagorean identity.

$- \cos^{2} (t^{2} h e a)$