To find the product of the matrices
[tex]\[
\left[\begin{array}{ll}
4 & 2
\end{array}\right]
\][/tex]
and
[tex]\[
\left[\begin{array}{cc}
-2 & 5 \\
7 & -1
\end{array}\right],
\][/tex]
we need to perform matrix multiplication according to the rules of linear algebra.
Here, we start with a [tex]\(1 \times 2\)[/tex] matrix and a [tex]\(2 \times 2\)[/tex] matrix. The result will be a [tex]\(1 \times 2\)[/tex] matrix.
Define the matrices:
[tex]\[
A = \left[\begin{array}{ll}
4 & 2
\end{array}\right]
\][/tex]
and
[tex]\[
B = \left[\begin{array}{cc}
-2 & 5 \\
7 & -1
\end{array}\right].
\][/tex]
To find the product [tex]\(AB\)[/tex], we perform element-wise multiplication and sum the results for each position.
The element in the first row and first column of the resulting matrix is calculated as follows:
[tex]\[
4 \cdot (-2) + 2 \cdot 7 = -8 + 14 = 6
\][/tex]
The element in the first row and second column is calculated as follows:
[tex]\[
4 \cdot 5 + 2 \cdot (-1) = 20 - 2 = 18
\][/tex]
Therefore, the resulting matrix after performing the multiplication is:
[tex]\[
\left[\begin{array}{ll}
6 & 18
\end{array}\right].
\][/tex]
Hence, the product of the given matrices is
[tex]\[
\left[\begin{array}{ll}
6 & 18
\end{array}\right].
\][/tex]
