Find matrix \( AB \) if
\[
A = \begin{bmatrix}
-1 & 2 & 3 \\
4 & -2 & 5
\end{bmatrix}, \quad
B = \begin{bmatrix}
2 & 1 \\
6 & 7 \\
5 & 3
\end{bmatrix}
\]
Find matrix \( AB \) if
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & -2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ 6 & 7 \\ 5 & 3 \end{bmatrix} \]
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Solution and Explanation
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:
\[ AB = \begin{bmatrix} 25 & 22 \\ 21 & 5 \end{bmatrix} \]
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