Question

Let $P$ and $Q$ be 2*2 matrices. Consider the statements.
$PQ=0\Rightarrow 1)P=0$ or $Q=0$ or $both$
2) $PQ=I_2\Rightarrow P=Q^{-1}$
3) $(P+Q{)}^2=P^2+2PQ+Q^2$

• A. $1$ and $2$ are false but $3$ is true
• B. $1$ and $3$ false and $2$ is true
• C. All are false
• D. All are true

Answer 1

The correct option is B $1$ and $3$ false and $2$ is true

1) Let $P=\left(\begin{array}{cc}1& 3\\ -1& -3\end{array}\right),Q=\left(\begin{array}{cc}3& -1\\ -1& \frac{1}{3}\end{array}\right)$
Now,$PQ=\left(\begin{array}{cc}1& 3\\ -1& -3\end{array}\right)\left(\begin{array}{cc}3& -1\\ -1& \frac{1}{3}\end{array}\right)$
$=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$
Here , $PQ=0$ but neither $P=0$ nor $Q=0$
2) $PQ=I_2$
$\Rightarrow \left(PQ\right)Q^{-1}={I_{2}Q}^{-1}$
$\Rightarrow P\left(Q{Q}^{-1}\right)=Q^{-1}$
$\Rightarrow P{I}_2=Q^{-1}$
$\Rightarrow P=Q^{-1}$
3) ${(P+Q)}^2=(P+Q)(P+Q)$
$=P^2+PQ+QP+Q^2$
In general matrix multiplication is not commutative i.e. $PQ\ne QP$
Hence, ${(P+Q)}^2\ne P^2+2PQ+Q^2$