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The problem asks you to find the matrix product \( AB \), where:

\[
A = \begin{bmatrix} -1 \\ 2 \\ -2 \\ 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 & 3 & 2 \end{bmatrix}
\]

To multiply these matrices, we perform the dot product between the columns of \( A \) and the rows of \( B \).

Given that:
- \( A \) is a 4x1 matrix (a column vector), and
- \( B \) is a 1x4 matrix (a row vector),

The result will be a 4x4 matrix. Here's how the multiplication is done for each entry of the result matrix:

\[
AB = \begin{bmatrix} -1 \\ 2 \\ -2 \\ 1 \end{bmatrix} \times \begin{bmatrix} 2 & 1 & 3 & 2 \end{bmatrix}
\]

Each element of the result matrix is computed by multiplying the element in \( A \) with every element in \( B \), as follows:

\[
AB = \begin{bmatrix}
-1 \times 2 & -1 \times 1 & -1 \times 3 & -1 \times 2 \\
2 \times 2 & 2 \times 1 & 2 \times 3 & 2 \times 2 \\
-2 \times 2 & -2 \times 1 & -2 \times 3 & -2 \times 2 \\
1 \times 2 & 1 \times 1 & 1 \times 3 & 1 \times 2
\end{bmatrix}
=
\begin{bmatrix}
-2 & -1 & -3 & -2 \\
4 & 2 & 6 & 4 \\
-4 & -2 & -6 & -4 \\
2 & 1 & 3 & 2
\end{bmatrix}
\]

The correct matrix is:

\[
AB = \begin{bmatrix} -2 & -1 & -3 & -2 \\ 4 & 2 & 6 & 4 \\ -4 & -2 & -6 & -4 \\ 2 & 1 & 3 & 2 \end{bmatrix}
\]

Thus, the correct answer is the third option:
\[
AB = \begin{bmatrix} -2 & -1 & -3 & -2 \\ 4 & 2 & 6 & 4 \\ -4 & -2 & -6 & -4 \\ 2 & 1 & 3 & 2 \end{bmatrix}
\]

Would you like more details, or do you have any questions about the steps?

Here are 5 related questions to expand your understanding:
1. What are the conditions for two matrices to be multiplied?
2. How do you compute the product of a 2x2 matrix and a 2x2 matrix?
3. What is the transpose of the matrix \( A \) in this problem?
4. What would happen if matrix \( B \) were 4x4? Could you still multiply them?
5. How does the result of matrix multiplication change if you reverse the order (i.e., \( BA \))?

**Tip:** Always check the dimensions of matrices before multiplying them to ensure the product is defined.

If possible, find AB where A = [[-1], [2], [-2], [1]] and B = [2, 1, 3, 2].

Solve the matrix multiplication problem for A = [[-1], [2], [-2], [1]] and B = [2, 1, 3, 2]. Learn the steps involved in multiplying a 4x1 matrix with a 1x4 matrix, producing a 4x4 matrix as a result.

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