The problem asks to calculate the matrix products $AB$ and $BA$, if possible, given the matrices:

$A = \begin{pmatrix} -1 & 0 \\ 2 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 \\ -1 & 1 \\ 0 & 2 \end{pmatrix}$

To solve this:

1. Dimensions of $A$: $A$ is a $2 \times 2$ matrix.
2. Dimensions of $B$: $B$ is a $3 \times 2$ matrix.

Step 1: Determine if $AB$ is possible.

For $AB$ to be defined, the number of columns in $A$ must equal the number of rows in $B$. But $A$ has 2 columns, while $B$ has 3 rows, so:

• **$AB$ is not possible because the dimensions do not match for matrix multiplication.

Step 2: Determine if $BA$ is possible.

For $BA$ to be defined, the number of columns in $B$ (which is 2) must equal the number of rows in $A$ (which is also 2). Since the number of columns of $B$ matches the number of rows of $A$, we can compute $BA$.

Now, I will calculate the matrix product $BA$.

$B = \begin{pmatrix} 3 & 1 \\ -1 & 1 \\ 0 & 2 \end{pmatrix}, \quad A = \begin{pmatrix} -1 & 0 \\ 2 & 4 \end{pmatrix}$

Performing the matrix multiplication $BA$:

$BA = \begin{pmatrix} 3 & 1 \\ -1 & 1 \\ 0 & 2 \end{pmatrix} \times \begin{pmatrix} -1 & 0 \\ 2 & 4 \end{pmatrix}$

Using the rules of matrix multiplication, we multiply each row of $B$ by each column of $A$.