This image shows two matrices that need to be multiplied together. Let's call them:

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

To find the product $AB$, we use matrix multiplication rules: the entry in row $i$, column $j$ of the resulting matrix $C$ is found by multiplying each element of row $i$ of $A$ with the corresponding element of column $j$ of $B$ and summing the results.

Let's calculate each element in the resulting matrix:

Step 1: Calculate Row 1

1. $c_{11} = (1 \cdot 3) + (2 \cdot 0) + (-1 \cdot -2) = 3 + 0 + 2 = 5$
2. $c_{12} = (1 \cdot 4) + (2 \cdot 1) + (-1 \cdot 0) = 4 + 2 + 0 = 6$
3. $c_{13} = (1 \cdot 2) + (2 \cdot 0) + (-1 \cdot 1) = 2 + 0 - 1 = 1$

Step 2: Calculate Row 2

1. $c_{21} = (3 \cdot 3) + (2 \cdot 0) + (0 \cdot -2) = 9 + 0 + 0 = 9$
2. $c_{22} = (3 \cdot 4) + (2 \cdot 1) + (0 \cdot 0) = 12 + 2 + 0 = 14$
3. $c_{23} = (3 \cdot 2) + (2 \cdot 0) + (0 \cdot 1) = 6 + 0 + 0 = 6$

Step 3: Calculate Row 3

1. $c_{31} = (-4 \cdot 3) + (0 \cdot 0) + (2 \cdot -2) = -12 + 0 - 4 = -16$
2. $c_{32} = (-4 \cdot 4) + (0 \cdot 1) + (2 \cdot 0) = -16 + 0 + 0 = -16$
3. $c_{33} = (-4 \cdot 2) + (0 \cdot 0) + (2 \cdot 1) = -8 + 0 + 2 = -6$

Resulting Matrix

Combining all of these, we get:

$AB = \begin{pmatrix} 5 & 6 & 1 \\ 9 & 14 & 6 \\ -16 & -16 & -6 \end{pmatrix}$

Would you like further explanation on any of these steps?

Here are five related questions to deepen understanding:

1. What would the result be if we switched the order of the matrices (i.e., $BA$)?
2. How is matrix multiplication different from element-wise multiplication?
3. Can you explain why the matrix product is not commutative in general?
4. What properties must two matrices satisfy to be conformable for multiplication?
5. How would you calculate the determinant of matrix $AB$ given $A$ and $B$?

Tip: For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.