question_answer

Show by an example that for \[A\ne O\] and \[B\ne O,\] \[AB=O.\]

Answer:

Let \[A=\left[ \begin{matrix}    0 & 1  \\    0 & 1  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    1 & 1  \\    0 & 0  \\ \end{matrix} \right]\] Clearly, \[A\ne O\] and \[B\ne O\] Now, \[AB=\left[ \begin{matrix}    0 & 1  \\    0 & 1  \\ \end{matrix} \right]\,\,\left[ \begin{matrix}    1 & 1  \\    0 & 0  \\ \end{matrix} \right]=\left[ \begin{matrix}    0 & 0  \\    0 & 0  \\ \end{matrix} \right]\]             = O Hence proved.