Question

Let $P$ and $Q$ be two $2\times 2$ matrices. Then the identity $(P+Q)(P-Q)=P^2-Q^2$ is

• A. True if $det\left(P\right)=det\left(Q\right)$
• B. True if $P$ or $Q$ is the identify matrix
• C. True if $P$ or $Q$ is the zero matrix
• D. True if $PQ=QP$

Answer 1

Answer: (B), (C), (D)

The correct options are
B True if $P$ or $Q$ is the zero matrix
C True if $P$ or $Q$ is the identify matrix
D True if $PQ=QP$

From the given information we know that, $P$ and $Q$ are two $2\times 2$ matrices.

The given identity is $(P+Q)(P-Q)=P^2-Q^2$ ...$\left(1\right)$

Now let's either $P$ or $Q$ is a zero matrix.

Then we can see the Identity given in equation $\left(1\right)$ satisfies truly. $($left-hand side of equation $1$ becomes equal to right-hand side.$)$

Now by Solving the left-hand side, we get $P^2-PQ+QP+Q^2$

$\Rightarrow P^2-PQ+QP-Q^2=P^2-Q^2$

$\Rightarrow PQ=QP$ ....$\left(2\right)$

So $PQ=QP$ then the given Identity satisfies truly too,

But if $PQ\ne QP$ and if one of $P$ and $Q$ is an Identity matrix then equation $\left(2\right)$ satisfies truly. $($left-hand side of equation $2$ becomes equal to right-hand side.$)$

So the given Identity satisfies truly if one of the matrix from $P$ and $Q$ is zero or Identity matrix or $PQ=QP$

Hence option $B,C$ and $D$ are correct.