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Let P be a non-singular matrix and $I + P + P^{2} + .... + P^{n} = 0$ then $P^{- 1}$ is

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$P^{n}$

$P$

$P^{n - 1}$

$I$

answer is A.

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Detailed Solution

$P + P^{2} + P^{3} + .... + P^{n} = - I P^{- 1} P + P^{- 1} P^{2} + P^{- 1} P^{3} + ... + P^{- 1} P^{n} = - P^{- 1} I I + P + P^{2} + ..... + P^{n - 1} = - P^{- 1} \Rightarrow - P^{n} = - P^{- 1} \Rightarrow P^{- 1} = P^{n}$

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