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 $\theta$)
= cos²$\theta$ + sin²$\theta$
= 1 ... [$\because$ cos²$\theta$ + sin²$\theta$ = 1]
$\therefore$ The determinant of the above matrix is 1.

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