https://file.aiquickdraw.com/mathbot/file/17300752729462j6nwuou.jpg

To solve this problem, we need to multiply the two matrices provided. Here’s a breakdown of the dimensions and steps involved:

1. **Matrix Dimensions**:
   - The first matrix \( A \) has dimensions \( 4 \times 2 \).
   - The second matrix \( B \) has dimensions \( 2 \times 3 \).
   
   Since the number of columns in \( A \) matches the number of rows in \( B \), matrix multiplication is possible, and the resulting matrix will have dimensions \( 4 \times 3 \).

2. **Multiplication Process**:
   To find the entry in row \( i \) and column \( j \) of the resulting matrix, multiply each element in row \( i \) of \( A \) by the corresponding element in column \( j \) of \( B \), then sum these products.

Let’s calculate each entry step-by-step:

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### Step-by-Step Calculation

Let \( A = \begin{bmatrix} 0 & -1 \\ -4 & 2 \\ 0 & 0 \\ -3 & -1 \\ 1 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & -4 & -5 \\ -5 & 4 & -1 \end{bmatrix} \).

#### Row 1 of \( A \):
\[
\begin{align*}
C_{11} &= (0)(-1) + (-1)(-5) = 0 + 5 = 5, \\
C_{12} &= (0)(-4) + (-1)(4) = 0 - 4 = -4, \\
C_{13} &= (0)(-5) + (-1)(-1) = 0 + 1 = 1.
\end{align*}
\]

#### Row 2 of \( A \):
\[
\begin{align*}
C_{21} &= (-4)(-1) + (2)(-5) = 4 - 10 = -6, \\
C_{22} &= (-4)(-4) + (2)(4) = 16 + 8 = 24, \\
C_{23} &= (-4)(-5) + (2)(-1) = 20 - 2 = 18.
\end{align*}
\]

#### Row 3 of \( A \):
\[
\begin{align*}
C_{31} &= (0)(-1) + (0)(-5) = 0 + 0 = 0, \\
C_{32} &= (0)(-4) + (0)(4) = 0 + 0 = 0, \\
C_{33} &= (0)(-5) + (0)(-1) = 0 + 0 = 0.
\end{align*}
\]

#### Row 4 of \( A \):
\[
\begin{align*}
C_{41} &= (-3)(-1) + (-1)(-5) = 3 + 5 = 8, \\
C_{42} &= (-3)(-4) + (-1)(4) = 12 - 4 = 8, \\
C_{43} &= (-3)(-5) + (-1)(-1) = 15 + 1 = 16.
\end{align*}
\]

#### Row 5 of \( A \):
\[
\begin{align*}
C_{51} &= (1)(-1) + (4)(-5) = -1 - 20 = -21, \\
C_{52} &= (1)(-4) + (4)(4) = -4 + 16 = 12, \\
C_{53} &= (1)(-5) + (4)(-1) = -5 - 4 = -9.
\end{align*}
\]

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### Final Result
Thus, the product \( C = AB \) is:
\[
C = \begin{bmatrix} 5 & -4 & 1 \\ -6 & 24 & 18 \\ 0 & 0 & 0 \\ 8 & 8 & 16 \\ -21 & 12 & -9 \end{bmatrix}.
\]

Would you like further explanation on matrix multiplication steps?

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### Related Questions to Explore
1. How can we determine if two matrices can be multiplied based on their dimensions?
2. What are the properties of matrix multiplication, such as distributive and associative properties?
3. How does the identity matrix function in matrix multiplication?
4. How does changing the order of multiplication affect the result in matrix multiplication?
5. How can matrix multiplication be used to represent transformations in geometry?

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### Tip
When multiplying matrices, always ensure the inner dimensions match (columns of the first matrix and rows of the second) to determine if multiplication is possible.

Calculate the product of the matrices:

\[
\begin{bmatrix} 0 & -1 \\ -4 & 2 \\ 0 & 0 \\ -3 & -1 \\ 1 & 4 \end{bmatrix} \times \begin{bmatrix} -1 & -4 & -5 \\ -5 & 4 & -1 \end{bmatrix}
\]

This solution shows how to multiply a 4x2 matrix by a 2x3 matrix, covering key concepts in linear algebra, such as matrix multiplication rules and step-by-step calculation. Suitable for high school students in Grades 10-12.

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