Consider the matrices.

B=[−4205] and C=[0141−14]

What is product matrix ​BC​ ?

We are asked to find the product of matrices \( B \) and \( C \).

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

Matrix \( B \) is a \( 4 \times 1 \) matrix, and matrix \( C \) is a \( 2 \times 3 \) matrix.

In this case, the multiplication of \( B \) and \( C \) is **not defined**. The number of columns in \( B \) (which is 1) must be equal to the number of rows in \( C \) (which is 2) for matrix multiplication to be valid, but they are not.

Hence, the product matrix \( BC \) does not exist due to dimension mismatch.

Would you like to discuss matrix multiplication properties, or details about this specific example?

Here are 5 related questions for further exploration:
1. What are the necessary conditions for matrix multiplication to be defined?
2. How do you find the dimensions of a product matrix?
3. Can matrices of different dimensions be multiplied in any special cases?
4. How does scalar multiplication differ from matrix multiplication?
5. What are some real-world applications of matrix multiplication?

**Tip**: Always check matrix dimensions before attempting to multiply them to ensure compatibility.

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

Explore why matrix multiplication BC is undefined for the matrices B=[−4 2 0 5] and C=[0 1 4 1 −1 4]. Understand the rules of matrix multiplication, the importance of dimension matching, and learn how to verify compatibility. Ideal for students in Grades 9-12.

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