Question

Suppose $A$ is any $3\times 3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$, If $\alpha A+\beta A^{-1}=4I$, then $\alpha +\beta$ is equal to :

• A. $8$
• B. $7$
• C. $13$
• D. $12$

Answer 1

The correct option is A $8$

$\begin{array}{c}\left(A-3I\right)\left(A-5I\right)=0\\ \Rightarrow A^2-8A+15I=0\\ \Rightarrow A-8I+15A^{-1}=0\\ \Rightarrow \frac{A}{2}+\frac{15}{2}{A}^{-1}=4I\\ \alpha =\frac{1}{2}=\beta =\frac{15}{2}\\ hence,\\ \alpha +\beta =8\end{array}$

hence, option $A$ is correct.