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The problem requires determining the matrix product \( AB \), where:

\[
A = \begin{bmatrix} -1 & 4 & 1 \\ -1 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ -2 \\ -5 \end{bmatrix}
\]

### Steps:
The product \( AB \) is found by performing the dot product of each row of \( A \) with the column vector \( B \).

#### First row of \( A \) with \( B \):
\[
(-1) \cdot 2 + 4 \cdot (-2) + 1 \cdot (-5) = -2 - 8 - 5 = -15
\]

#### Second row of \( A \) with \( B \):
\[
(-1) \cdot 2 + 1 \cdot (-2) + 1 \cdot (-5) = -2 - 2 - 5 = -9
\]

Thus, the product \( AB \) is:

\[
AB = \begin{bmatrix} -15 \\ -9 \end{bmatrix}
\]

Would you like further explanation or details on matrix multiplication? 

Here are some related questions you might find interesting:
1. What are the properties of matrix multiplication?
2. How do you verify the dimensions of matrices before multiplication?
3. What happens if you multiply a matrix by the identity matrix?
4. How do matrix operations apply in solving systems of linear equations?
5. Can matrix multiplication be performed in a different order?

**Tip:** Always check the dimensions of matrices before multiplying. The number of columns in the first matrix must match the number of rows in the second matrix.

Determine the product AB where A = [[-1, 4, 1], [-1, 1, 1]] and B = [[2], [-2], [-5]].

Determine the product of matrices A and B, where A is a 2x3 matrix and B is a 3x1 matrix. This step-by-step solution explains the multiplication process and the final result. Ideal for high school students studying linear algebra and matrix operations.

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