Question

State whether the following statement is True or False: If matrix $AB=O,$ then $A=O$ or $B=O$ or both 𝐴 and 𝐵 are null matrices

• A. True
• B. False

Answer 1

The correct option is B False

Let A=$\left[\begin{array}{ccc}1& 0& \\ 0& 0& \end{array}\right]$and B=$\left[\begin{array}{ccc}0& 0& \\ 0& 1& \end{array}\right]$
Hence,

AB=$\left[\begin{array}{ccc}1& 0& \\ 0& 0& \end{array}\right]\times \left[\begin{array}{ccc}0& 0& \\ 0& 1& \end{array}\right]=\left[\begin{array}{ccc}0& 0& \\ 0& 0& \end{array}\right]=0$
Thus $A,B\ne O$ but $AB=O$
Hence, the given statement is False.