Question

If the matrix AB = O, then

• A. A = O or B = O
• B. A = O and B = O
• C. It is not necessary that either A = O or B = O
• D. $A \neq O, B \neq O$

Answer 1

Answer: (C)

It is not necessary that either A = O or B = O

Answer 2

$AB = O \Rightarrow \left|AB \right| = 0 \Rightarrow \left|A\right|. \left|B\right|=0$ $\Rightarrow \left|A\right| =0 \,or \,\left|B\right| = 0$ when AB = O, neither A nor B may be O. For example if $A = \begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$ and $B = \begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$ , then $AB = \begin{bmatrix}1&0\\\ 0&0\end{bmatrix} \begin{bmatrix}0&0\\\ 1&0\end{bmatrix} = \begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$ But none of A and B are zero matrices. So if AB is zero it is not necessary that either A = O or B =O.