Topic: Mathematics
The eigen values of the matrix $\begin{bmatrix}cos~\alpha & sin~\alpha \\ sin~\alpha & cos~\alpha\end{bmatrix}$ are
- A. $\pm cos~\alpha$
- B. $\pm sin~\alpha$
- C. $tan~\alpha$ & $cot~\alpha$
- D. $cos~\alpha \pm sin~\alpha$
Answer: D. $cos~\alpha \pm sin~\alpha$
Explanation: Let $A=\begin{bmatrix}cos~\alpha & sin~\alpha \\ sin~\alpha & cos~\alpha\end{bmatrix}$. Eigen values are given as $|sI-A|=0 \Rightarrow \begin{vmatrix}s-cos~\alpha & -sin~\alpha \\ -sin~\alpha & s-cos~\alpha\end{vmatrix}=0 \Rightarrow (s-cos~\alpha)^{2}-sin^{2}\alpha=0 \Rightarrow (s-cos~\alpha)=\pm sin~\alpha \Rightarrow s=cos~\alpha\pm sin~\alpha$.