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)

$I - A + A^2$

Answer 2

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, $(I+ A)^{-1}=I - A+ A^2$ .