Consider the matrices.

B=[−4025] and C=[0140−14]

What is product matrix ​BC​ ?

To find the product matrix \( BC \), we need to perform matrix multiplication. Given matrices are:

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

Matrix \( B \) is a \( 2 \times 2 \) matrix, and matrix \( C \) is a \( 2 \times 3 \) matrix. The product \( BC \) will be a \( 2 \times 3 \) matrix.

### Step-by-Step Multiplication

To find each element of the product matrix \( BC \), we use the formula:

\[
(BC)_{ij} = \sum_{k=1}^{2} B_{ik} \cdot C_{kj}
\]

1. **Element at position (1,1)**:
   \[
   (BC)_{11} = (-4 \cdot 0) + (0 \cdot 0) = 0
   \]

2. **Element at position (1,2)**:
   \[
   (BC)_{12} = (-4 \cdot 1) + (0 \cdot -1) = -4
   \]

3. **Element at position (1,3)**:
   \[
   (BC)_{13} = (-4 \cdot 4) + (0 \cdot 4) = -16
   \]

4. **Element at position (2,1)**:
   \[
   (BC)_{21} = (2 \cdot 0) + (5 \cdot 0) = 0
   \]

5. **Element at position (2,2)**:
   \[
   (BC)_{22} = (2 \cdot 1) + (5 \cdot -1) = 2 - 5 = -3
   \]

6. **Element at position (2,3)**:
   \[
   (BC)_{23} = (2 \cdot 4) + (5 \cdot 4) = 8 + 20 = 28
   \]

### Product Matrix \( BC \):

\[
BC = \begin{bmatrix} 0 & -4 & -16 \\ 0 & -3 & 28 \end{bmatrix}
\]

### Conclusion:

The resulting matrix after multiplication is:

\[
\boxed{\begin{bmatrix} 0 & -4 & -16 \\ 0 & -3 & 28 \end{bmatrix}}
\]

Would you like any further details or explanations? 

### Related Questions:
1. What are the requirements for two matrices to be multiplied?
2. How do you determine the size of the resulting matrix after multiplication?
3. What are some properties of matrix multiplication (e.g., associativity, distributivity)?
4. How does element-wise matrix multiplication differ from the standard matrix product?
5. Can you perform matrix multiplication with non-square matrices?

### Tip:
Always ensure that the number of columns in the first matrix matches the number of rows in the second matrix to be able to multiply them.

Consider the matrices B = [-4 0; 2 5] and C = [0 1 4; 0 -1 4]. What is the product matrix BC?

Learn how to perform matrix multiplication for matrices B = [-4 0; 2 5] and C = [0 1 4; 0 -1 4] with a detailed step-by-step solution. This problem covers key concepts in linear algebra and matrix multiplication, suitable for students in Grades 10-12.

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