Question

If $A$ is a square matrix.such that $A^3=0$, then $(I+A{)}^{-1}$ is

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

Answer 1

Answer: (C)

We have, $A^3=0$
$\Rightarrow I+A^3=I\Rightarrow (I+A)(I-A+A^2)=I$
$\Rightarrow I-A+A^2=(I+A{)}^{-1}$
Hence, $(I+A{)}^{-1}=I-A+A^2$ .