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

Answer: (A)

Given $(A-3 I)(A-5 I)=0$
$\Rightarrow A^{2}-8 A+15 I=0 $
$\Rightarrow A-8 I+15 A^{-1}=0 $
$\Rightarrow \frac{1}{2} A+\frac{15}{2} A^{-1}=4 I$
Given $\,\,\,\alpha A+\beta A ^{-1}=4 I$
So, $\,\,\,\alpha=\frac{1}{2}, \beta=\frac{15}{2}$
Therefore, $\,\,\,\alpha+\beta=\frac{16}{2}=8$