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

MCQ

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
$I+2^n P$
✓
$I+\left(2^n-1\right) P$

✓

Answer

Correct option: D.

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

d
(d)

Given, $\quad P^2=P$

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

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

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

$=2^n I$

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

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

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

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