Single Choice

Let $$P$$ and $$Q$$ be 2*2 matrices. Consider the statements. $$PQ=0 \Rightarrow 1) P=0$$ or $$Q=0$$ or $$\mathrm{both}$$ 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
Correct Answer
CAll are false
DAll are true

Solution

1) Let $$P=\begin{pmatrix} 1 & 3 \\ -1 & -3 \end{pmatrix},Q=\begin{pmatrix} 3 & -1 \\ -1 & \frac { 1 }{ 3 } \end{pmatrix}$$
Now,$$ PQ=\begin{pmatrix} 1 & 3 \\ -1 & -3 \end{pmatrix}\begin{pmatrix} 3 & -1 \\ -1 & \frac { 1 }{ 3 } \end{pmatrix}$$

$$=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$
Here , $$PQ=0$$ but neither $$P=0$$ nor $$Q=0$$

2) $$PQ=I_{2}$$
$$\Rightarrow (PQ){ Q }^{ -1 }={ I_{ 2 }Q }^{ -1 }$$
$$\Rightarrow P(Q{ Q }^{ -1 })={ Q }^{ -1 }$$
$$\Rightarrow P{I}_{2}={ Q }^{ -1 }$$
$$\Rightarrow P={ Q }^{ -1 }$$

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, $${ (P+Q) }^{ 2 }\neq { P }^{ 2 }+2PQ+{ Q }^{ 2 }$$

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