The question asks to find the matrix $A^3$ for the given matrix $A$. Let's first extract the matrix $A$ and compute $A^3$.

Matrix $A$ is given as:

$A = \begin{bmatrix} 3 & 1 \\ 2 & -2 \end{bmatrix}$

To calculate $A^3$, we first find $A^2$, then multiply that result by $A$ again.

Let me calculate that for you.The matrix $A^3$ is:

$A^3 = \begin{bmatrix} 35 & 9 \\ 18 & -10 \end{bmatrix}$

This matches option D from the choices provided.

Do you need any further details or clarification?

Here are 5 related questions:

1. How do we calculate the determinant of a 2x2 matrix?
2. What is the inverse of matrix $A$, if it exists?
3. How can you use eigenvalues and eigenvectors to compute powers of matrices?
4. What is the trace of a matrix, and what is its significance?
5. How does matrix multiplication differ from scalar multiplication?

Tip: When multiplying matrices, always remember that matrix multiplication is not commutative (i.e., $AB \neq BA$ in general).