To determine which of the given options represents the product of the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to perform matrix multiplication.
Given:
[tex]\[
A = \begin{pmatrix}
1 & -4 \\
-3 & 2 \\
7 & -5
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
1 & 7 \\
-2 & 11
\end{pmatrix}
\][/tex]
To multiply matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we use the definition of matrix multiplication where the element at the [tex]\( i \)[/tex]-th row and [tex]\( j \)[/tex]-th column of the product matrix is the dot product of the [tex]\( i \)[/tex]-th row of [tex]\( A \)[/tex] and the [tex]\( j \)[/tex]-th column of [tex]\( B \)[/tex].
Let’s compute each element step-by-step:
First Row:
1. [tex]\((1 \cdot 1) + (-4 \cdot -2) = 1 + 8 = 9\)[/tex]
2. [tex]\((1 \cdot 7) + (-4 \cdot 11) = 7 - 44 = -37\)[/tex]
So, the first row of [tex]\( AB \)[/tex] is [tex]\([9, -37]\)[/tex].
Second Row:
1. [tex]\((-3 \cdot 1) + (2 \cdot -2) = -3 - 4 = -7\)[/tex]
2. [tex]\((-3 \cdot 7) + (2 \cdot 11) = -21 + 22 = 1\)[/tex]
So, the second row of [tex]\( AB \)[/tex] is [tex]\([-7, 1]\)[/tex].
Third Row:
1. [tex]\((7 \cdot 1) + (-5 \cdot -2) = 7 + 10 = 17\)[/tex]
2. [tex]\((7 \cdot 7) + (-5 \cdot 11) = 49 - 55 = -6\)[/tex]
So, the third row of [tex]\( AB \)[/tex] is [tex]\([17, -6]\)[/tex].
Now compiling all rows together, the product matrix [tex]\( AB \)[/tex] is:
[tex]\[
AB = \begin{pmatrix}
9 & -37 \\
-7 & 1 \\
17 & -6
\end{pmatrix}
\][/tex]
Comparing this with the given options, we see that the correct answer is:
B. [tex]\( AB = \begin{pmatrix}
9 & -37 \\
-7 & 1 \\
17 & -6
\end{pmatrix}
\)[/tex]
