Question

# Mark each of the following statement true or false: If A and B are matrices such that AB = O and A neq O, then B = O.

Matrices
Mark each of the following statement true or false: If A and B are matrices such that AB = O and $$A \neq O$$, then B = O.

2021-01-14
Step 1
Given statement,
If A and B are matrices such that AB = O and $$A \neq O$$, then B = O.
Step 2
Take A and B to be two matrices
$$A=\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}\neq 0 \text{ and } B=\begin{bmatrix}0 & 0 \\1 & 0 \end{bmatrix}$$
$$AB=\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}\begin{bmatrix}0 & 0 \\1 & 0 \end{bmatrix}$$
$$AB=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix} \text{ and } A=\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix} \neq 0$$
But B is non Zero.
Therefore, the given statement is false.