To solve the matrix multiplication problem, we need to multiply row vector [tex]\([4, 2]\)[/tex] with the [tex]\(2 \times 2\)[/tex] matrix [tex]\(\left[\begin{array}{cc}-2 & 5 \\ 7 & -1\end{array}\right]\)[/tex].
Here are the steps to perform this multiplication:
1. Multiply each element of the row vector by the corresponding column in the matrix and sum the results.
2. For the [tex]\(1^{st}\)[/tex] element of the resulting vector:
[tex]\[
(4 \times -2) + (2 \times 7)
\][/tex]
3. For the [tex]\(2^{nd}\)[/tex] element of the resulting vector:
[tex]\[
(4 \times 5) + (2 \times -1)
\][/tex]
Now performing the computations:
[tex]\[
4 \times -2 = -8
\][/tex]
[tex]\[
2 \times 7 = 14
\][/tex]
[tex]\[
-8 + 14 = 6
\][/tex]
This gives us the [tex]\(1^{st} \)[/tex] element of the resulting vector.
Next:
[tex]\[
4 \times 5 = 20
\][/tex]
[tex]\[
2 \times -1 = -2
\][/tex]
[tex]\[
20 - 2 = 18
\][/tex]
This gives us the [tex]\(2^{nd} \)[/tex] element of the resulting vector.
Hence, the resulting vector when multiplying [tex]\([4, 2]\)[/tex] with [tex]\(\left[\begin{array}{cc}-2 & 5 \\ 7 & -1\end{array}\right]\)[/tex] is:
[tex]\[
\left[\begin{array}{c}6 \\ 18\end{array}\right]
\][/tex]
Therefore, the correct product is:
[tex]\[
\left[\begin{array}{c}6 \\ 18\end{array}\right]
\][/tex]
