Let P be a non-singular matrix such that I+P+P2+ ldots ..+Pn= O (where O denotes the null matrix), then P- 1 is

Question

Let P be a non-singular matrix such that I+P+P2+ ldots ..+Pn= O (where O denotes the null matrix), then P- 1 is (A) Pn (B) - Pn (C) - (1 + P + P2 + ldots . + Pn) (D) None of these

Tardigrade Admin · Accepted Answer

Given, I+P+P2+ ldots ldots +Pn= O P- 1+(P- 1 P)+(P- 1 P)P+ ldots ldots .+(P- 1 P)Pn - 1=O ⇒ P- 1+ I+IP+ ldots ldots ..+ I Pn - 1= O ⇒ P- 1= Pn

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

$P^{n}$

$- P^{n}$

$- (1 + P + P^{2} + \dots . + P^{n})$

None of these

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

Q. Let P be a non-singular matrix such that I+P+P2+…..+Pn=O (where O denotes the null matrix), then P−1 is

Pn

−Pn

−(1+P+P2+….+Pn)