Given an example of two non-zero 2 × 2 matrices A and B such that AB = O.

Question:

Given an example of two non-zero $2 \times 2$ matrices $A$ and $B$ such that $A B=O$.

Solution:

Let the two matrices be $A=\left[\begin{array}{ll}5 & 0 \\ 9 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & 0 \\ 12 & -21\end{array}\right]$, such that $A B=\left[\begin{array}{ll}5 & 0 \\ 9 & 0\end{array}\right]\left[\begin{array}{cc}0 & 0 \\ 12 & -21\end{array}\right]$.

$\Rightarrow A B=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

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