Let p be a non-singular matrix, and $I +p+p^2+...+p^n = O$. Then find $p^{-1}$.

$-p$ $p$ $p^n$ none of these

$p^n$

We have, $I +p+p^2+...+p^n = O$   ...(1) Since p is non-singular matrix, p is invertible. Multiplying both sides of (1) by $p^{-1}$, we get $⇒p^{-1}I + p^{-1}p + p^{-1}p^2 + ... + p^{-1}p^n= p^{-1}O$ $⇒p^{-1}+I+p+p^2+...+p^{n-1}=O$ $⇒p^{-1}=-(I+p+ p^2+...+p^{n-1})$ $⇒p^{-1}=-(-p^n) =p^n$