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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 \( A \).

\[
BA = \begin{pmatrix} 
(3 \times -1) + (1 \times 2) & (3 \times 0) + (1 \times 4) \\ 
(-1 \times -1) + (1 \times 2) & (-1 \times 0) + (1 \times 4) \\ 
(0 \times -1) + (2 \times 2) & (0 \times 0) + (2 \times 4)
\end{pmatrix}
\]

Simplifying:

\[
BA = \begin{pmatrix} 
-3 + 2 & 0 + 4 \\ 
1 + 2 & 0 + 4 \\ 
0 + 4 & 0 + 8
\end{pmatrix}
\]

\[
BA = \begin{pmatrix} 
-1 & 4 \\ 
3 & 4 \\ 
4 & 8
\end{pmatrix}
\]

### Conclusion:
- \( AB \) is not defined.
- \( BA = \begin{pmatrix} -1 & 4 \\ 3 & 4 \\ 4 & 8 \end{pmatrix} \).

Would you like more details on this calculation or have any further questions? Here are some follow-up questions:

1. How can we generalize the rule for determining if matrix multiplication is possible?
2. What are the conditions for matrices to be invertible?
3. What happens when the dimensions of the matrices for multiplication don’t align?
4. How does matrix multiplication differ from regular scalar multiplication?
5. How can we interpret the resulting matrix in a real-world context?

**Tip:** When multiplying matrices, always ensure the inner dimensions match. The result will have the outer dimensions.

Apskaičiuokite AB ir BA, jei galima, kai A = \begin{pmatrix} -1 & 0 \\ 2 & 4 \end{pmatrix}, B = \begin{pmatrix} 3 & 1 \\ -1 & 1 \\ 0 & 2 \end{pmatrix}.

Learn how to calculate the matrix products AB and BA, given the matrices A = \begin{pmatrix} -1 & 0 \\ 2 & 4 \end{pmatrix} and B = \begin{pmatrix} 3 & 1 \\ -1 & 1 \\ 0 & 2 \end{pmatrix}. This example covers essential matrix multiplication rules in linear algebra and demonstrates when these products are possible.

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