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.

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.