To determine the product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to perform matrix multiplication. Given the matrices:
[tex]\[
A = \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]
\quad \text{and} \quad
B = \left[\begin{array}{cc}-1 & 2 \\ 1 & 0\end{array}\right],
\][/tex]
we will calculate the product matrix [tex]\( AB = C \)[/tex].
Matrix [tex]\( C \)[/tex] will also be a [tex]\( 2 \times 2 \)[/tex] matrix, and to find each entry [tex]\( c_{ij} \)[/tex] of matrix [tex]\( C \)[/tex], we use the dot product of the [tex]\( i \)[/tex]-th row of [tex]\( A \)[/tex] with the [tex]\( j \)[/tex]-th column of [tex]\( B \)[/tex].
1. For the entry [tex]\( c_{11} \)[/tex] (first row, first column):
[tex]\[
c_{11} = (1 \cdot -1) + (2 \cdot 1) = -1 + 2 = 1
\][/tex]
2. For the entry [tex]\( c_{12} \)[/tex] (first row, second column):
[tex]\[
c_{12} = (1 \cdot 2) + (2 \cdot 0) = 2 + 0 = 2
\][/tex]
3. For the entry [tex]\( c_{21} \)[/tex] (second row, first column):
[tex]\[
c_{21} = (3 \cdot -1) + (4 \cdot 1) = -3 + 4 = 1
\][/tex]
4. For the entry [tex]\( c_{22} \)[/tex] (second row, second column):
[tex]\[
c_{22} = (3 \cdot 2) + (4 \cdot 0) = 6 + 0 = 6
\][/tex]
Thus, the product matrix [tex]\( AB = C \)[/tex] is:
[tex]\[
C = \left[\begin{array}{cc}1 & 2 \\ 1 & 6\end{array}\right]
\][/tex]
Therefore, the correct product matrix is:
[tex]\[
\left[\begin{array}{cc}1 & 2 \\ 1 & 6\end{array}\right]
\][/tex]
So, the correct answer is option D:
[tex]\[
\left[\begin{array}{ll}1 & 2 \\ 1 & 6\end{array}\right]
\][/tex]
