Question

Find matrix \( AB \) if

\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & -2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ 6 & 7 \\ 5 & 3 \end{bmatrix} \]

Hint:

When multiplying matrices, ensure the number of columns in the first matrix matches the number of rows in the second matrix.

Answer 1

Step 1: Matrix multiplication formula for \( AB \) is:

\[ AB = \begin{bmatrix} -1 & 2 & 3 \\ 4 & -2 & 5 \end{bmatrix} \times \begin{bmatrix} 2 & 1 \\ 6 & 7 \\ 5 & 3 \end{bmatrix} \]

Step 2: Perform matrix multiplication for each element:

• First row, first column: \( (-1)(2) + (2)(6) + (3)(5) = -2 + 12 + 15 = 25 \)
• First row, second column: \( (-1)(1) + (2)(7) + (3)(3) = -1 + 14 + 9 = 22 \)
• Second row, first column: \( (4)(2) + (-2)(6) + (5)(5) = 8 - 12 + 25 = 21 \)
• Second row, second column: \( (4)(1) + (-2)(7) + (5)(3) = 4 - 14 + 15 = 5 \)

Thus, the resulting matrix \( AB \) is:

\[ AB = \begin{bmatrix} 25 & 22 \\ 21 & 5 \end{bmatrix} \]