Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$, where $I=I_3$ and $O=O_3$. If…
Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+$ $\beta A^{-1}=4 I$, then $\alpha+\beta$ is equal to
8
12
13
7
Solution
We have
$
\begin{aligned}
&(A-3 I)(A-5 I)=O \\
&\Rightarrow A^2-8 A+15 I=O
\end{aligned}
$
Multiplying both sides by $A^{-1}$, we get;
$
\begin{aligned}
A^{-1} A \cdot A-8 A^{-1} A+15 A^{-1} I=A^{-1} O \\
\Rightarrow & A-8 I+15 A^{-1}=O \\
& A+15 A^{-1}=8 I \\
& \frac{A}{2}+\frac{15 A^{-1}}{2}=4 I \\
\therefore & \alpha+\beta=\frac{1}{2}+\frac{15}{2}=\frac{16}{2}=8
\end{aligned}
$