Question

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Explanation:

Step1: Check matrix dimensions

First matrix: \( 2 \times 4 \) (2 rows, 4 columns). Second matrix: \( 2 \times 2 \) (2 rows, 2 columns). For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \( 4
eq 2 \), so multiplication is not defined. Wait, no—wait, first matrix: rows = 2, columns = 4. Second matrix: rows = 2, columns = 2. Wait, no, the first matrix is \(

$$\begin{bmatrix} -5 & 0 & 4 & 1 \\ 5 & 0 & 4 & 4 \end{bmatrix}$$

\) (2 rows, 4 columns). Second matrix: \(

$$\begin{bmatrix} -4 & 5 \\ 0 & 1 \end{bmatrix}$$

\) (2 rows, 2 columns). The rule is: if matrix \( A \) is \( m \times n \) and matrix \( B \) is \( p \times q \), then \( A \times B \) is defined only if \( n = p \). Here, \( n = 4 \), \( p = 2 \), so \( 4
eq 2 \). Wait, no—wait, maybe I mixed up. Wait, first matrix: columns = 4, second matrix: rows = 2. So 4 ≠ 2, so product is not defined? Wait, no, wait the second matrix: let's check again. Wait, the second matrix is \(

$$\begin{bmatrix} -4 & 5 \\ 0 & 1 \end{bmatrix}$$

\), which is 2 rows, 2 columns. The first matrix is 2 rows, 4 columns. So when multiplying \( A \times B \), \( A \) is \( m \times n \), \( B \) is \( p \times q \), so \( n \) (columns of A) must equal \( p \) (rows of B). Here, \( n = 4 \), \( p = 2 \), so 4 ≠ 2. Therefore, the product is not defined? Wait, no, wait maybe I made a mistake. Wait, no—wait, the first matrix is \( 2 \times 4 \), second is \( 2 \times 2 \). So columns of A (4) ≠ rows of B (2). So multiplication is not defined. Wait, but wait, maybe the second matrix is transposed? No, the problem shows the second matrix as \(

$$\begin{bmatrix} -4 & 5 \\ 0 & 1 \end{bmatrix}$$

\). So, the product is not defined? Wait, no, wait the first matrix: rows = 2, columns = 4. Second matrix: rows = 2, columns = 2. So for \( A \times B \), the number of columns of A must equal the number of rows of B. So 4 (columns of A) vs 2 (rows of B) → not equal. So product is not defined. Wait, but wait, maybe I flipped the matrices? If it's \( B \times A \), but the problem is \( A \times B \). So the product is not defined. Wait, but let's recheck. Wait, the first matrix: \( 2 \times 4 \), second: \( 2 \times 2 \). So \( n = 4 \), \( p = 2 \). So \( 4
eq 2 \), so multiplication is undefined.

Step2: Confirm the rule

Matrix multiplication rule: \( A_{m \times n} \times B_{p \times q} \) is defined only if \( n = p \). Here, \( n = 4 \), \( p = 2 \), so not defined.

Answer:

B. The product is not defined.