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 $P^{3}=P\left(\right.I-P\left.\right)=PI-P^{2}$
$=PI-\left(\right.I-P\left.\right)$
$=P-I+P$
$=2P-I$
$P^{4}=P\cdot P^{3}=P\left(\right.2P-I\left.\right)$
$\Rightarrow P^{4}=2P^{2}-P$
$\Rightarrow P^{4}=2I-2P-P$
$\Rightarrow P^{4}=2I-3P$
$P^{5}=P\left(\right.2I-3P\left.\right)$
$\Rightarrow P^{5}=2P-3\left(\right.I-P\left.\right)$
$\Rightarrow P^{5}=5P-3I$
$P^{6}=P\left(\right.5P-3I\left.\right)=5P^{2}-3P$
$=5\left(\right.I-P\left.\right)-3P$
$\Rightarrow P^{6}=5I-8P$
$\therefore n=6$