Question

Calculate the matrix product \( C = AB \), where
\[A = \begin{bmatrix} 5 & 3 & 1 \\ 4 & 6 & 2 \\ 10 & 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 5 \\ 2 & 4 \\ 3 & 2 \end{bmatrix}\]

• A. \(\begin{bmatrix} 14 & 38 \\ 22 & 48 \\ 28 & 70 \end{bmatrix}\)
• B. \(\begin{bmatrix} 14 & 39 \\ 22 & 48 \\ 29 & 70 \end{bmatrix}\)
• C. \(\begin{bmatrix} 14 & 39 \\ 22 & 48 \\ 28 & 70 \end{bmatrix}\)
• D. \(\begin{bmatrix} 14 & 39 \\ 22 & 49 \\ 28 & 70 \end{bmatrix}\)

Hint:

When performing matrix multiplication, make sure you multiply corresponding elements and then sum them to get each element of the resulting matrix.

Answer 1

The Correct Option is C

To find the matrix product \( C = AB \), we perform the following matrix multiplication: 

\[C = \begin{bmatrix} 5 & 3 & 1 \\ 4 & 6 & 2 \\ 10 & 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 5 \\ 2 & 4 \\ 3 & 2 \end{bmatrix}\]

 The product is computed as follows: \[ C_{11} = 5(1) + 3(2) + 1(3) = 5 + 6 + 3 = 14 \] \[ C_{12} = 5(5) + 3(4) + 1(2) = 25 + 12 + 2 = 39 \] \[ C_{21} = 4(1) + 6(2) + 2(3) = 4 + 12 + 6 = 22 \] \[ C_{22} = 4(5) + 6(4) + 2(2) = 20 + 24 + 4 = 48 \] \[ C_{31} = 10(1) + 3(2) + 4(3) = 10 + 6 + 12 = 28 \] \[ C_{32} = 10(5) + 3(4) + 4(2) = 50 + 12 + 8 = 70 \] Thus, the matrix \( C \) is: 

\[C = \begin{bmatrix} 14 & 39 \\ 22 & 48 \\ 28 & 70 \end{bmatrix}\]