Question

For a square matrix $P$ satisfying the relation $P^2=I-P$ , where $I$ is an identity matrix, and if $P^n=5I-8P$ then the value of $n$ is:

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

Answer: (C)

Now $P^3=P(I-P)=PI-P^2$
$=PI-(I-P)$
$=P-I+P$
$=2P-I$
$P^4=P\cdot P^3=P(2P-I)$
$\Rightarrow P^4=2P^2-P$
$\Rightarrow P^4=2I-2P-P$
$\Rightarrow P^4=2I-3P$
$P^5=P(2I-3P)$
$\Rightarrow P^5=2P-3(I-P)$
$\Rightarrow P^5=5P-3I$
$P^6=P(5P-3I)=5P^2-3P$
$=5(I-P)-3P$
$\Rightarrow P^6=5I-8P$
$\therefore n=6$