To solve the problem of finding the product of the given matrix and vector, we will perform matrix multiplication. Here are the detailed steps:
We are given a matrix:
[tex]\[
\begin{bmatrix}
4 & -2 \\
-3 & 5 \\
9 & 10 \\
6 & 11
\end{bmatrix}
\][/tex]
and a vector:
[tex]\[
\begin{bmatrix}
7 \\
-4
\end{bmatrix}
\][/tex]
We need to find the product of the matrix and the vector. The result will be a new vector obtained by performing the dot product of each row of the matrix with the vector.
Step-by-step, we will calculate each element of the resulting vector:
1. First element:
[tex]\[
(4 \times 7) + (-2 \times -4) = 28 + 8 = 36
\][/tex]
2. Second element:
[tex]\[
(-3 \times 7) + (5 \times -4) = -21 - 20 = -41
\][/tex]
3. Third element:
[tex]\[
(9 \times 7) + (10 \times -4) = 63 - 40 = 23
\][/tex]
4. Fourth element:
[tex]\[
(6 \times 7) + (11 \times -4) = 42 - 44 = -2
\][/tex]
Thus, the product of the given matrix and vector is:
[tex]\[
\begin{bmatrix}
36 \\
-41 \\
23 \\
-2
\end{bmatrix}
\][/tex]
So the final result is:
[tex]\[
\begin{array}{c}
{\left[\begin{array}{cc}
4 & -2 \\
-3 & 5 \\
9 & 10 \\
6 & 11
\end{array}\right]\left[\begin{array}{c}
7 \\
-4
\end{array}\right]} = \\
{[[\begin{array}{c}
36 \\
-41 \\
23 \\
-2
\end{array}]]}
\end{array}
\][/tex]
