Question

If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\mathrm{and}A\left(\mathrm{adj}A=\right)\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$, then find the value of k.

Answer 1

$$\begin{gathered} A=\left[\cos \theta \sin \theta \\ -\sin \theta \cos \theta \right] \\ \therefore \left|A\right|=\left|\cos \theta \sin \theta \\ -\sin \theta \cos \theta \right|={\cos }^2\theta +{\sin }^2\theta =1\ne 0 \\ \mathrm{Thus},A^{-1}\mathrm{exists}. \\ \mathrm{Now}, \\ {A}^{-1}=\frac{\mathrm{adj}A}{\left|A\right|}=\left[\cos \theta -\sin \theta \\ \sin \theta \cos \theta \right] \\ \Rightarrow A^{-1}=\mathrm{adj}A \\ \Rightarrow A{A}^{-1}=A\mathrm{adj}A \\ \Rightarrow A{A}^{-1}=\left[k0 \\ 0k\right] \\ \Rightarrow \left[10 \\ 01\right]=\left[k0 \\ 0k\right][\because A{A}^{-1}=I] \\ \Rightarrow k=1 \end{gathered}$$