Question

For a square matrix $A,$ if $A^3=O,$ then $I+A+A^2$ is equal to

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

Answer 1

The correct option is D $(I-A{)}^{-1}$

$A^3=O$
$\Rightarrow I-A^3=I$
$\Rightarrow (I-A)(I+A+A^2)=I$
$\Rightarrow I-A$ and $I+A+A^2$ are inverse of each other.
$\therefore I+A+A^2=(I-A{)}^{-1}$