Question

If $P$ is involutory matrix and $Q$ is idempotent matrix and $PQ=P,$ then $P{Q}^{100}+P^{100}Q=$

• A. $I$
• B. $I+QP$
• C. $PQ$
• D. $PQ+I$

Answer 1

The correct option is D $PQ+I$

Given : $P$ is involutory
$\Rightarrow P^{2k}=I$ and $P^{2k+1}=P,k\in N$
and for $Q$ to be idempotent : $Q^n=Q,n\ge 2,n\in N$
Now,
$P{Q}^{100}+P^{100}Q=PQ[P^{99}+Q^{99}]=PQ[P+Q]=\left(PQ\right)P+P{Q}^2=P^2+PQ=I+PQ$