# Determinant of $\left[\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right]$

The calculator will find the determinant of the square $3$x$3$ matrix $\left[\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right]$, with steps shown.

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Calculate $\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right|$.

### Solution

Subtract column $1$ multiplied by $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$ from column $2$: $C_{2} = C_{2} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} C_{1}$.

$\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = \left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & 0 & - 4 \sin{\left(2 x \right)}\end{array}\right|$

Subtract column $1$ multiplied by $\frac{\sin{\left(2 x \right)}}{\cos{\left(x \right)}}$ from column $3$: $C_{3} = C_{3} - \frac{\sin{\left(2 x \right)}}{\cos{\left(x \right)}} C_{1}$.

$\left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & 0 & - 4 \sin{\left(2 x \right)}\end{array}\right| = \left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & 0\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & 0 & - 3 \sin{\left(2 x \right)}\end{array}\right|$

Expand along row $1$:

$\left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & 0\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & 0 & - 3 \sin{\left(2 x \right)}\end{array}\right| = \left(\cos{\left(x \right)}\right) \left(-1\right)^{1 + 1} \left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 2} \left|\begin{array}{cc}- \sin{\left(x \right)} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & - 3 \sin{\left(2 x \right)}\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 3} \left|\begin{array}{cc}- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}}\\- \cos{\left(x \right)} & 0\end{array}\right| = \cos{\left(x \right)} \left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right|$

The determinant of a 2x2 matrix is $\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$.

$\left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right| = \left(\frac{1}{\cos{\left(x \right)}}\right)\cdot \left(- 3 \sin{\left(2 x \right)}\right) - \left(2 \cos^{2}{\left(x \right)}\right)\cdot \left(0\right) = - 6 \sin{\left(x \right)}$

Finally, $\left(\cos{\left(x \right)}\right)\cdot \left(- 6 \sin{\left(x \right)}\right) = - 3 \sin{\left(2 x \right)}$.

$\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = - 3 \sin{\left(2 x \right)}$A