Let P be an m × m matrix such that P^2=P . Then, (I+P)^n equals

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3 and 4 .Determinants and Matrices

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Let $P$ be an $m \times m$ matrix such that $P^2=P$. Then, $(I+P)^n$ equals

A

$I+P$

B

$I+n P$

C

$I+2^n P$

D

$I+\left(2^n-1\right) P$

(KVPY-2011)

Solution

(d)

Given, $\quad P^2=P$

$(I+P)^n=(I+I)^n$

$\left[\because P^2=P \Rightarrow P^{-1} P^2=P^{-1} P=P=I\right]$

$\Rightarrow \quad(I+P)^n=(2 I)^n$

$=2^n I$

$=\left(2^n-1+1\right) I$

$=I+\left(2^n-1\right) I$

$=I+\left(2^n-1\right) P \quad[\because I=P]$

Standard 12

Mathematics

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