Question

Consider the following statements:
1. The product of two non-zero matrices can never be identity matrix.
2. The product of two non-zero matrices can never be zero matrix.
Which of the above statements is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (D)

Neither 1 nor 2

Product of a matrix and it's inverse is always an identity matrix.

Example :-

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

And both are non-zero.

Thus, statement $1$ is incorrect.

Product of two non-zero matrices can be a zero matrix. Example,

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

Thus, statement $2$ is incorrect.

Hence, option D is correct.