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, $PQ=Q \ldots\left(i\right)$
and $QP=P \ldots\left(ii\right)$
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 $Q^{2}=Q]$