Question

Let $P$ be a non-singular matrix such that $I+P+P^{2}+\ldots ..+P^{n}= \, O$ (where $O$ denotes the null matrix), then $P^{- 1}$ is

• A. $P^{n}$
• B. $- \, P^{n}$
• C. $- \, \left(1 + P + P^{2} + \, \ldots . + \, P^{n}\right)$
• D. None of these

Answer 1

Answer: (A)

Given, $I+P+P^{2}+ \, \ldots \ldots +P^{n}= \, O$
$P^{- 1}+\left(P^{- 1} P\right)+\left(P^{- 1} P\right)P+ \, \ldots \ldots .+\left(P^{- 1} P\right)P^{n - 1}=O$
$\Rightarrow \, P^{- 1}+ \, I+IP+\ldots \ldots ..+ \, I P^{n - 1}= \, O$
$\Rightarrow \, P^{- 1}= \, P^{n}$