Question

If $A$ is a non-singular matrix such that $(A-2I)(A-4I)=O$, where $I$ is an identity matrix of the same order as that of $A$. Then $\frac{1}{6}A+\frac{4}{3}{A}^{-1}$ is equal to:

• A. $O$
• B. $I$
• C. $2I$
• D. $6I$

Answer 1

The correct option is B $I$

We have,
$(A-2I)(A-4I)=O$
$\Rightarrow A^2-2A-4A+8I=O$
$\Rightarrow A^2-6A+8I=O$
$\Rightarrow A^{-1}(A^2-6A+8I)=A^{-1}O$
$\Rightarrow A-6I+8A^{-1}=O$
$\Rightarrow A+8A^{-1}=6I$
$\Rightarrow \frac{1}{6}A+\frac{4}{3}{A}^{-1}=I$