Question

If the matrix $AB$ is a zero matrix, then which one of the following is correct?

• A. $A$ must be equal to zero matrix or $B$ must be equal to zero matrix
• B. $A$ must be equal to zero matrix and $B$ must be equal to zero matrix
• C. It is not necessary that either $A$ is zero matrix or $B$ is zero matrix
• D. None of the above

Answer 1

Answer: (C)

It is not necessary that either $A$ is zero matrix or $B$ is zero matrix

Consider the matrices $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]and\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]BotharenonzeromatrixButtheproductiszeromatrix$
So option C is correct