https://file.aiquickdraw.com/mathbot/file/1726426263963ei0yz25s.jpg

To find the product of these two matrices, we'll follow the matrix multiplication rules. Let's define the matrices as follows:

\[
A = \begin{bmatrix} 1 & -2 & 3 \\ 4 & 0 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -2 \\ 1 & 0 \\ 2 & -4 \end{bmatrix}
\]

Since matrix \(A\) is a \(2 \times 3\) matrix and matrix \(B\) is a \(3 \times 2\) matrix, the resulting matrix \(C = A \times B\) will be a \(2 \times 2\) matrix. The elements of the resulting matrix \(C\) are computed using the formula:

\[
C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}
\]

Where:
- \(i\) is the row index in matrix \(A\),
- \(j\) is the column index in matrix \(B\),
- \(n\) is the number of columns in \(A\) (or rows in \(B\)).

Let's calculate each element of the resulting matrix \(C\):

1. **Element \(C_{11}\):**
   \[
   C_{11} = (1 \times 0) + (-2 \times 1) + (3 \times 2) = 0 - 2 + 6 = 4
   \]

2. **Element \(C_{12}\):**
   \[
   C_{12} = (1 \times -2) + (-2 \times 0) + (3 \times -4) = -2 + 0 - 12 = -14
   \]

3. **Element \(C_{21}\):**
   \[
   C_{21} = (4 \times 0) + (0 \times 1) + (6 \times 2) = 0 + 0 + 12 = 12
   \]

4. **Element \(C_{22}\):**
   \[
   C_{22} = (4 \times -2) + (0 \times 0) + (6 \times -4) = -8 + 0 - 24 = -32
   \]

So, the product of the matrices \(A\) and \(B\) is:

\[
C = A \times B = \begin{bmatrix} 4 & -14 \\ 12 & -32 \end{bmatrix}
\]

Would you like a more detailed explanation of any part of this process or have any questions?

### Related Questions
1. How do you determine the dimensions of the resulting matrix when multiplying two matrices?
2. What conditions must be met for matrix multiplication to be possible?
3. Can matrix multiplication be performed if the number of columns in the first matrix does not equal the number of rows in the second matrix?
4. How does matrix multiplication differ from element-wise matrix multiplication?
5. How do you calculate the inverse of a matrix, and when does a matrix have no inverse?

**Tip:** Always ensure that the number of columns in the first matrix matches the number of rows in the second matrix when performing matrix multiplication.

Find the product of the following two matrices:
Matrix A: [[1, -2, 3], [4, 0, 6]]
Matrix B: [[0, -2], [1, 0], [2, -4]]

Learn how to calculate the product of two matrices, a 2x3 matrix and a 3x2 matrix. The solution involves step-by-step matrix multiplication following the formula C_ij = sum(A_ik * B_kj). Perfect for students in grades 10-12 studying linear algebra.

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