That is, if $A$ is a $2 \times 2$ matrix, then $A^2 = AA$.
Now, we want to find a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$. This means that $AA = 0$, where $0$ is the $2 \times 2$ zero matrix.
One example of such a matrix is:
$$
A = \begin{pmatrix}
1
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