If \( A=\begin{bmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{bmatrix} \), then \( A^2-(2\cos\alpha)A \) is equal to:
(Where I is identity matrix of order 2)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( -\mathrm{I} \)
Every 2 × 2 matrix satisfies the Cayley-Hamilton relation
\( A^2-(\mathrm{tr}\,A)A+|A|\,I=O \)
Here
\( \mathrm{tr}\,A=2\cos\alpha \)
\( |A|=\cos^2\alpha+\sin^2\alpha=1 \)
So
\( A^2-(2\cos\alpha)A+I=O \)
\( A^2-(2\cos\alpha)A=-I \)