Question

Let $A=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right),B=\left(\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right),i=\sqrt{-1}$
$\text{Statement-I}:(A+B)(A-B)=A^2-B^2\text{Statement-II}:AB=BA$
Then which of the following is correct regarding the given statements:

• A. Statement $1$ is false. statement $2$ is true.
• B. Statement $1$ is true. statement $2$ is true ; statement $2$ is correct explanation for statement $1$
• C. Statement $1$ is true. statement $2$ is true ; statement $2$ is not a correct explanation for statment $1$.
• D. Statement $1$ is true. statement $2$ is false.

Answer 1

The correct option is B Statement $1$ is true. statement $2$ is true ; statement $2$ is correct explanation for statement $1$

$AB=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)\left(\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

$BA=\left(\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right)\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
Clearly, we have: $AB=BA$
$\therefore$ Statement -$2$ is true
Now $(A+B)(A-B)=A^2-AB+BA-B^2=A^2-B^2$
Hence Statement $1$ is true, and statement $2$ is correct explanation for statement $1$