The problem asks to find the result of $A^2 \cdot B^T$, where matrices $A$ and $B$ are given, and $B^T$ is the transpose of matrix $B$. Let's break it down:

Given matrices: $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 1 \\ 2 & -2 \\ -3 & 3 \end{pmatrix}$

1. Step 1: Compute $A^2$

   $A^2$ is found by multiplying matrix $A$ by itself: $A^2 = A \cdot A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ Perform matrix multiplication to get $A^2$.

2. Step 2: Compute the transpose of $B$

   The transpose of matrix $B$, denoted $B^T$, is obtained by swapping rows with columns: $B^T = \begin{pmatrix} -1 & 2 & -3 \\ 1 & -2 & 3 \end{pmatrix}$

3. Step 3: Multiply $A^2$ with $B^T$

   After finding $A^2$ and $B^T$, multiply them together: $A^2 \cdot B^T$ Perform matrix multiplication to get the result.

I will now calculate these steps.