https://file.aiquickdraw.com/mathbot/file/1729740366460s1974irq.jpg

Let's solve the matrix multiplication problem shown in the image.

We are asked to find the product of the following two matrices:

\[
A = \begin{bmatrix} 3 & 2 & 0 \\ 0 & 5 & 1 \\ -2 & 2 & -3 \end{bmatrix}, 
B = \begin{bmatrix} -1 & 3 & 0 \\ 1 & 1 & -4 \\ 1 & 1 & 1 \end{bmatrix}
\]

The product of two matrices \( A \) (with dimensions \(m \times n\)) and \( B \) (with dimensions \(n \times p\)) results in a matrix \( C \) with dimensions \(m \times p\), where each element \( C_{ij} \) is calculated as:

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

We multiply the rows of matrix \( A \) by the columns of matrix \( B \).

### Step-by-step calculation:
1. **Row 1 of \(A\) multiplied by Columns of \(B\):**
   - \(C_{11} = (3 \times -1) + (2 \times 1) + (0 \times 1) = -3 + 2 + 0 = -1\)
   - \(C_{12} = (3 \times 3) + (2 \times 1) + (0 \times 1) = 9 + 2 + 0 = 11\)
   - \(C_{13} = (3 \times 0) + (2 \times -4) + (0 \times 1) = 0 - 8 + 0 = -8\)

2. **Row 2 of \(A\) multiplied by Columns of \(B\):**
   - \(C_{21} = (0 \times -1) + (5 \times 1) + (1 \times 1) = 0 + 5 + 1 = 6\)
   - \(C_{22} = (0 \times 3) + (5 \times 1) + (1 \times 1) = 0 + 5 + 1 = 6\)
   - \(C_{23} = (0 \times 0) + (5 \times -4) + (1 \times 1) = 0 - 20 + 1 = -19\)

3. **Row 3 of \(A\) multiplied by Columns of \(B\):**
   - \(C_{31} = (-2 \times -1) + (2 \times 1) + (-3 \times 1) = 2 + 2 - 3 = 1\)
   - \(C_{32} = (-2 \times 3) + (2 \times 1) + (-3 \times 1) = -6 + 2 - 3 = -7\)
   - \(C_{33} = (-2 \times 0) + (2 \times -4) + (-3 \times 1) = 0 - 8 - 3 = -11\)

### Result:
\[
C = \begin{bmatrix} -1 & 11 & -8 \\ 6 & 6 & -19 \\ 1 & -7 & -11 \end{bmatrix}
\]

Let me know if you want more details or have any questions.

Here are 5 related questions to expand on this topic:
1. How do you verify if two matrices can be multiplied?
2. What are the dimensions of the resulting matrix after multiplying two matrices?
3. How do you calculate the determinant of a 3x3 matrix?
4. What happens if you multiply a matrix by an identity matrix?
5. How can matrix multiplication be applied in solving systems of linear equations?

**Tip:** Always check the dimensions of the matrices before attempting to multiply them!

Find the product of the following two matrices: [[3, 2, 0], [0, 5, 1], [-2, 2, -3]] and [[-1, 3, 0], [1, 1, -4], [1, 1, 1]]

Learn how to perform matrix multiplication with two 3x3 matrices: [[3, 2, 0], [0, 5, 1], [-2, 2, -3]] and [[-1, 3, 0], [1, 1, -4], [1, 1, 1]]. Detailed step-by-step calculations show the product of the matrices. Suitable for Grades 10-12.

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