Question

Let P and Q be two $2\times 2$ matrices. Consider the statements i)$PQ=O\Rightarrow P=O$ or $Q=O$ or both
ii) $PQ=I_2\Rightarrow P=Q^{-1}$
iii) ${(P+Q)}^2=P^2+2PQ+Q^2$. Then

• A. (i) and (ii) are false while (iii) is true
• B. (i) and (iii) are false while (ii) is true
• C. (ii) and (iii) are false while (i) is true
• D. None

Answer 1

Answer: (B)

(i) and (iii) are false while (ii) is true

In matrices $PQ=0$ doesn't means that $P=0$ and $Q=0$

$⟹1)$ is false

$PQ=I$

$PQ{Q}^{-1}=I{Q}^{-1}⟹PI=Q^{-1}⟹P=Q^{-1}$

$2)$ is true

$(P+Q{)}^2=(P+Q\left)\right(P+Q)=P^2+PQ+QP+Q^2$

$3)$ is false $(\because$ matrix multiplication is not commutative i.e. $PQ\ne QP)$