Question

If A=[2 0 6] and \(B=\begin{bmatrix} 3&\ \ 5\\7&-2\\6&\ \ 6 \end{bmatrix}\), then AB=

• A. [42 46]
• B. \(\begin{bmatrix} 42\\46 \end{bmatrix}\)
• C. \(B=\begin{bmatrix} 6&10\\0&0\\36&36 \end{bmatrix}\)
• D. [17 19]
• E. \(\begin{bmatrix} 2&12\\14&-4\end{bmatrix}\)

Answer 1

The Correct Option is A

We are asked to calculate the product \( AB \). The matrix \( A \) is a 1x3 matrix and \( B \) is a 3x2 matrix. The product \( AB \) will be a 1x2 matrix. To calculate the elements of the product, perform the matrix multiplication: \[ AB = \begin{bmatrix} 2 & 0 & 6 \end{bmatrix} \times \begin{bmatrix} 3 & 5 \\ 7 & -2 \\ 6 & 6 \end{bmatrix}. \] For the first element of the resulting matrix: \[ (2 \times 3) + (0 \times 7) + (6 \times 6) = 6 + 0 + 36 = 42. \] For the second element of the resulting matrix: \[ (2 \times 5) + (0 \times -2) + (6 \times 6) = 10 + 0 + 36 = 46. \] Thus, the product matrix is \[ AB = \begin{bmatrix} 42 & 46 \end{bmatrix}. \]

Answer 2

We are given two matrices:

\[ A = \begin{bmatrix} 2 & 0 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 5 \\ 7 & -2 \\ 6 & 6 \end{bmatrix} \] We need to compute the matrix product \( AB \). Note that \( A \) is a 1x3 matrix, and \( B \) is a 3x2 matrix. The resulting product \( AB \) will be a 1x2 matrix. To perform the matrix multiplication, we calculate each element of the resulting matrix by taking the dot product of the row of \( A \) with the corresponding column of \( B \). 1. First element: \[ 2 \times 3 + 0 \times 7 + 6 \times 6 = 6 + 0 + 36 = 42 \] 2. Second element: \[ 2 \times 5 + 0 \times (-2) + 6 \times 6 = 10 + 0 + 36 = 46 \] Thus, the matrix product \( AB \) is: \[ AB = \begin{bmatrix} 42 & 46 \end{bmatrix} \] .