Since $\mathbf{A}^3 = \mathbf{0},$ the eigenvalues of $\mathbf{A}$ must be 0.  Then by the Cayley-Hamilton Theorem, $\mathbf{A}^2 = \mathbf{0}.$  Thus, there is only $\boxed{1}$ possibility for $\mathbf{A}^2.$
The answer is: 1
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$  Find the number of different possible matrices that $\mathbf{A}^2$ can be.  If you think the answer is infinite, then enter "infinite".

### Answer:
