Question

Let $P$ be a non-singular matrix and $I+P+P^2+\cdots +P^n=O$. Then $P^{-1}$ is equal to (where $n\in Z^{+}$)

• A. $P^n$
• B. $P$
• C. $P^{n-1}$
• D. $I$

Answer 1

The correct option is A $P^n$

$I+P+P^2+\cdots +P^n=O$
on multipling by $P^{-1}$ on both sides, we get
$\Rightarrow P^{-1}+I+P+\cdots +P^{n-1}=O$
$\Rightarrow P^{-1}=-(I+P+\cdots +P^{n-1})$
$=-(-P^n)=P^n$