To find the product of the matrices [tex]\(\mathbf{A}\)[/tex] and [tex]\(\mathbf{B}\)[/tex], we need to perform matrix multiplication. Let's denote the matrices as follows:
[tex]\[
\mathbf{A} = \begin{pmatrix}
4 & -2 \\
-3 & 5 \\
9 & 10 \\
6 & 11
\end{pmatrix}, \;
\mathbf{B} = \begin{pmatrix}
7 \\
-4
\end{pmatrix}
\][/tex]
We need to determine the resulting matrix [tex]\(\mathbf{C} = \mathbf{A} \cdot \mathbf{B}\)[/tex]. Matrix [tex]\(\mathbf{C}\)[/tex] will have the same number of rows as [tex]\(\mathbf{A}\)[/tex] and the same number of columns as [tex]\(\mathbf{B}\)[/tex]. Hence, [tex]\(\mathbf{C}\)[/tex] will be a 4x1 matrix.
Perform the multiplication row by row:
1. First row calculation:
[tex]\[
C_{1} = (4 \cdot 7) + (-2 \cdot -4) = 28 + 8 = 36
\][/tex]
2. Second row calculation:
[tex]\[
C_{2} = (-3 \cdot 7) + (5 \cdot -4) = -21 - 20 = -41
\][/tex]
3. Third row calculation:
[tex]\[
C_{3} = (9 \cdot 7) + (10 \cdot -4) = 63 - 40 = 23
\][/tex]
4. Fourth row calculation:
[tex]\[
C_{4} = (6 \cdot 7) + (11 \cdot -4) = 42 - 44 = -2
\][/tex]
By combining all these row results, we obtain the final product matrix [tex]\(\mathbf{C}\)[/tex]:
[tex]\[
\mathbf{C} = \begin{pmatrix}
36 \\
-41 \\
23 \\
-2
\end{pmatrix}
\][/tex]
Thus, the product of the given matrices [tex]\(\mathbf{A}\)[/tex] and [tex]\(\mathbf{B}\)[/tex] is:
[tex]\[
\left[\begin{array}{cc}
4 & -2 \\
-3 & 5 \\
9 & 10 \\
6 & 11
\end{array}\right]\left[\begin{array}{c}
7 \\
-4
\end{array}\right]
=
\left[\begin{array}{c}
36 \\
-41 \\
23 \\
-2
\end{array}\right]
\][/tex]
