If $A$ is a non-zero square matrix of order $n$ with $\operatorname{det}(I+A) \neq 0$ and $A^3=O$, where $I,…
If $A$ is a non-zero square matrix of order $n$ with $\operatorname{det}(I+A) \neq 0$ 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
$I-A+A^2$
$I+A+A^2$
$I+A^{-1}$
$I+A$
Solution
Given, $|I+A| \neq O$
ie, $(A+I)$ is a non-singular matrix.
$O \rightarrow$ Null Matrix
$I \rightarrow$ Unit Matrix
$\because \quad I^3=I$
$\Rightarrow \quad A^3=0$
$\Rightarrow \quad A^3+I=O+I$
$\Rightarrow \quad A^3+I^3=O+I$
$\Rightarrow \quad(A+I)\left(A^2-A+I\right)=(O+I)$
$(A+I)\left(A^2-A+I\right)=I \quad \because(O+I=I)$
Operate $(A+I)^{-1}$ on both sides
$\left\{(A+I)^{-1}(A+I)\right\}\left(A^2-A+I\right)$
$=(A+I)^{-1} \cdot I$
$\Rightarrow \quad I \cdot\left(A^2-A+I\right)=(A+I)^{-1}$
$\left(\because I \cdot(A+I)^{-1}=(A+I)^{-1}\right)$
$\Rightarrow \quad(A+I)^{-1}=\left(A^2-A+I\right)$
or
$(I+A)^{-1}=\left(I-A+A^2\right)$