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. $13$
• B. $7$
• C. $12$
• D. $8$

Answer 1

The correct option is D $8$

$(A-3I)(A-5I)=O$
$\Rightarrow A^2-8AI+15=0$
$\Rightarrow A-8I+15A^{-1}=0$
$\Rightarrow A+15A^{-1}=8I$
$\Rightarrow \frac{1}{2}A+\frac{15}{2}{A}^{-1}=4I$
$\Rightarrow \alpha =\frac{1}{2}$ and $\beta =\frac{15}{2}$
Hence, $\alpha +\beta =8$