Let p be a non-singular matrix, and $I +p+p^2+...+p^n = O$. Then find $p^{-1}$.
Answer & explanation
Correct answer: option 3
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
$⇒p^{-1}I + p^{-1}p + p^{-1}p^2 + ... + p^{-1}p^n= p^{-1}O$
$⇒p^{-1}+I+p+p^2+...+p^{n-1}=O$
$⇒p^{-1}=-(I+p+ p^2+...+p^{n-1})$
$⇒p^{-1}=-(-p^n) =p^n$