Question

Let $P$ and $Q$ are matrices such that $PQ=Q$ and $QP=P$ , then $P^2+Q^2=$

• A. $P$
• B. $Q$
• C. $P+Q$
• D. $P-Q$

Answer 1

Answer: (C)

We have, $PQ=Q\dots \left(i\right)$
and $QP=P\dots \left(ii\right)$
From $\left(i\right)$, $PQ=Q$
$\Rightarrow Q\left(PQ\right)=Q\cdot Q$
$\Rightarrow \left(QP\right)Q=Q^2$
$\Rightarrow PQ=Q^2$
$\Rightarrow Q=Q^2$ [using (ii)]
From $\left(ii\right)$, $QP=P$
$\Rightarrow P\left(QP\right)=P\cdot P$
$\Rightarrow \left(PQ\right)P=P^2$
$\Rightarrow QP=P^2$
$\Rightarrow P=P^2$
[using(ii)]
Now, $P^2+Q^2=P+Q$
$[\because P^2=P$ and $Q^2=Q]$