Question

If A = $\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]$, then for any natural number n, find the value of Det(A").

Answer 1

Given A = $\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]$ $\Rightarrow A^n=\left[\begin{array}{cc}cosn\theta & sinn\theta \\ -sinn\theta & cosn\theta \end{array}\right]$ $\Rightarrow \left|A^n\right|i.e.,Det\left(A^n\right)=\left[\begin{array}{cc}cosn\theta & sinn\theta \\ -sinn\theta & cosn\theta \end{array}\right]=co{s}^{2}n\theta +si{n}^{2}n\theta =1.$