Question

If $A$ is a non-zero square matrix of order $n$ with det $(I+A) \neq 0$ and $A^{3}=O$, where $I, O$ are unit and null matrices of order $n \times n$ respectively, then $(I+A)^{-1}$ is equal to

• A. $I-A+A^{2}$
• B. $I+A+A^{2}$
• C. $I+A^{-1}$
• D. I+A

Answer 1

Answer: (A)

Given, $|I+A| \neq O$
e, $(A+I)$ is a non-singular matrix.
$O \rightarrow$ Null Matrix
$I\rightarrow$ Unit Matrix
$\because I^{3}=I$
$\Rightarrow A^{3}=O$
$\Rightarrow A^{3}+I=O+I$
$\Rightarrow A^{3}+I^{3}=O+I$
$\Rightarrow (A+I)\left(A^{2}-A+I\right)=(O+I)$
$(A+I)\left(A^{2}-A+I\right)=I \, \because(O+I=I)$
Operate $(A+I)^{-1}$ on both sides
$\left\{(A+I)^{-1}(A+I)\right\}\left(A^{2}-A+I\right)$
$=(A+I)^{-1} \cdot I$
$\Rightarrow I \cdot\left(A^{2}-A+I\right)=(A+I)^{-1}$
$\left(\because I \cdot(A+I)^{-1}=(A+I)^{-1}\right)$
$(A+I)^{-1}=\left(A^{2}-A+I\right)$
or $(I+A)^{-1}=\left(I-A+A^{2}\right)$