# Determinant of $\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right]$

The calculator will find the determinant of the square $2$x$2$ matrix $\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right]$, with steps shown.

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Calculate $\left|\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right|$.

### Solution

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

$\left|\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right| = \left(\frac{\sqrt{3}}{2}\right)\cdot \left(- \sin{\left(t \right)}\right) - \left(\frac{\cos{\left(t \right)}}{2}\right)\cdot \left(0\right) = - \frac{\sqrt{3} \sin{\left(t \right)}}{2}$

$\left|\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right| = - \frac{\sqrt{3} \sin{\left(t \right)}}{2}\approx - 0.866025403784439 \sin{\left(t \right)}$A