Evaluate the determinant | cc & - & | - Vidyadip

Question

Evaluate the determinant $\left|\begin{array}{cc} {\cos \theta} & {-\sin \theta} \\ {\sin \theta} & {\cos \theta} \end{array}\right|$

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Answer

We know that determinant of $A$ is calculated as $|A|=\left|\begin{array}{ll} {a} & {b} \\ {c} & {d} \end{array}\right|= ad - bc$
Now, $\left|\begin{array}{cc} {\cos \theta} & {-\sin \theta} \\ {\sin \theta} & {\cos \theta} \end{array}\right|$
$= \cos \theta (\cos \theta ) - (-\sin \theta )(\sin \theta)$
$= \cos^2\theta + \sin^{2 }\theta$
$= 1 ... [\because \cos^{2 }\theta + \sin^{2 }\theta = 1]$
$\therefore$ The determinant of the above matrix is $1.$

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