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To find the product of the given matrices, we'll multiply each element in the rows of the first matrix by the corresponding element in the columns of the second matrix and then add them together.

The matrices are:
\[
A = \begin{bmatrix} -3 & 0 & -4 \\ -3 & 3 & -2 \\ 4 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 0 & 1 \\ -2 & 1 & -1 \\ 0 & 1 & -1 \end{bmatrix}
\]

The resulting product \( AB \) will be a \(3 \times 3\) matrix, where each entry \( c_{ij} \) is given by:
\[
c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj}
\]

Let's calculate each entry of \( AB \) step by step.

### First Row of the Product
1. **\( c_{11} \):**
   \[
   (-3)(-1) + (0)(-2) + (-4)(0) = 3 + 0 + 0 = 3
   \]

2. **\( c_{12} \):**
   \[
   (-3)(0) + (0)(1) + (-4)(1) = 0 + 0 - 4 = -4
   \]

3. **\( c_{13} \):**
   \[
   (-3)(1) + (0)(-1) + (-4)(-1) = -3 + 0 + 4 = 1
   \]

So, the first row of \( AB \) is \([3, -4, 1]\).

### Second Row of the Product
1. **\( c_{21} \):**
   \[
   (-3)(-1) + (3)(-2) + (-2)(0) = 3 - 6 + 0 = -3
   \]

2. **\( c_{22} \):**
   \[
   (-3)(0) + (3)(1) + (-2)(1) = 0 + 3 - 2 = 1
   \]

3. **\( c_{23} \):**
   \[
   (-3)(1) + (3)(-1) + (-2)(-1) = -3 - 3 + 2 = -4
   \]

So, the second row of \( AB \) is \([-3, 1, -4]\).

### Third Row of the Product
1. **\( c_{31} \):**
   \[
   (4)(-1) + (1)(-2) + (0)(0) = -4 - 2 + 0 = -6
   \]

2. **\( c_{32} \):**
   \[
   (4)(0) + (1)(1) + (0)(1) = 0 + 1 + 0 = 1
   \]

3. **\( c_{33} \):**
   \[
   (4)(1) + (1)(-1) + (0)(-1) = 4 - 1 + 0 = 3
   \]

So, the third row of \( AB \) is \([-6, 1, 3]\).

### Final Product Matrix
Putting it all together, we get:
\[
AB = \begin{bmatrix} 3 & -4 & 1 \\ -3 & 1 & -4 \\ -6 & 1 & 3 \end{bmatrix}
\]

Let me know if you'd like a detailed breakdown of the multiplication process or have any questions!

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Here are some related questions to explore further:

1. How do we determine if matrix multiplication between two matrices is possible?
2. What properties of matrices are important for matrix multiplication (e.g., commutativity, associativity)?
3. Can matrix multiplication be used to solve systems of linear equations?
4. How does matrix multiplication apply to transformations in 3D space?
5. What is the role of the identity matrix in matrix multiplication?

**Tip:** When calculating matrix products by hand, organizing calculations row by row and column by column can help avoid errors.

Find the product of the following two matrices:

\[ \begin{bmatrix} -3 & 0 & -4 \\ -3 & 3 & -2 \\ 4 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} -1 & 0 & 1 \\ -2 & 1 & -1 \\ 0 & 1 & -1 \end{bmatrix} \]

Calculate the product of two 3x3 matrices with a detailed solution. This problem covers the rules of matrix multiplication, using row-by-column calculations to find each element in the resulting matrix. Ideal for high school students in Grades 10-12 studying linear algebra.

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