To find the product matrix [tex]\( AB \)[/tex], we follow the matrix multiplication rule, where each element of the resulting matrix is the dot product of the corresponding row of matrix [tex]\( A \)[/tex] and the corresponding column of matrix [tex]\( B \)[/tex].
Given matrices:
[tex]\[ A = \begin{bmatrix} 1 & -4 \\ -3 & 2 \\ 7 & -5 \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 1 & 7 \\ -2 & 11 \end{bmatrix} \][/tex]
Let's perform the matrix multiplication step by step:
1. Calculate the element in the first row, first column of [tex]\( AB \)[/tex]:
[tex]\[ (1 \cdot 1) + (-4 \cdot -2) = 1 + 8 = 9 \][/tex]
2. Calculate the element in the first row, second column of [tex]\( AB \)[/tex]:
[tex]\[ (1 \cdot 7) + (-4 \cdot 11) = 7 - 44 = -37 \][/tex]
3. Calculate the element in the second row, first column of [tex]\( AB \)[/tex]:
[tex]\[ (-3 \cdot 1) + (2 \cdot -2) = -3 - 4 = -7 \][/tex]
4. Calculate the element in the second row, second column of [tex]\( AB \)[/tex]:
[tex]\[ (-3 \cdot 7) + (2 \cdot 11) = -21 + 22 = 1 \][/tex]
5. Calculate the element in the third row, first column of [tex]\( AB \)[/tex]:
[tex]\[ (7 \cdot 1) + (-5 \cdot -2) = 7 + 10 = 17 \][/tex]
6. Calculate the element in the third row, second column of [tex]\( AB \)[/tex]:
[tex]\[ (7 \cdot 7) + (-5 \cdot 11) = 49 - 55 = -6 \][/tex]
Thus, the resulting product matrix [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{bmatrix} 9 & -37 \\ -7 & 1 \\ 17 & -6 \end{bmatrix} \][/tex]
The correct answer is:
B. [tex]\( AB = \begin{bmatrix} 9 & -37 \\ -7 & 1 \\ 17 & -6 \end{bmatrix} \)[/tex]
