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

Concept:

 \(\rm A \neq 0\) and \(\rm B \neq 0\) But AB = 0.

Calculations:

Given,  the matrix AB is a zero matrix

⇒ AB = 0

Consider,  \(\rm A = \begin{bmatrix} 0 &0 \\ 5 & 0 \end{bmatrix}\)and \(\rm B = \begin{bmatrix} 0 &0 \\ 4 &0 \end{bmatrix}\)

⇒\(\rm A \neq 0\) and \(\rm B \neq 0\)

Now \(\rm AB = \begin{bmatrix} 0 &0 \\ 5 & 0 \end{bmatrix}.\begin{bmatrix} 0 &0 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}.\)

⇒AB = 0

Here, \(\rm A \neq 0\) and \(\rm B \neq 0\) but AB = 0.

Hence, if the matrix AB is a zero matrix, then It is not necessary that either A is zero matrix or B is zero matrix.