To determine the product of the matrices
[tex]\[
A = \begin{pmatrix} 1 & 3 & 1 \\ -2 & 1 & 0 \end{pmatrix}
\][/tex]
and
[tex]\[
B = \begin{pmatrix} 2 & -2 \\ 3 & 5 \\ 4 & 1 \end{pmatrix},
\][/tex]
let's follow the process of matrix multiplication step-by-step.
Matrix multiplication is carried out by taking the dot product of the rows of the first matrix with the columns of the second matrix.
1. First element of the product (position [1,1]):
Take the dot product of the first row of matrix [tex]\(A\)[/tex] with the first column of matrix [tex]\(B\)[/tex]:
[tex]\[
(1 \cdot 2) + (3 \cdot 3) + (1 \cdot 4) = 2 + 9 + 4 = 15
\][/tex]
2. Second element of the product (position [1,2]):
Take the dot product of the first row of matrix [tex]\(A\)[/tex] with the second column of matrix [tex]\(B\)[/tex]:
[tex]\[
(1 \cdot -2) + (3 \cdot 5) + (1 \cdot 1) = -2 + 15 + 1 = 14
\][/tex]
3. Third element of the product (position [2,1]):
Take the dot product of the second row of matrix [tex]\(A\)[/tex] with the first column of matrix [tex]\(B\)[/tex]:
[tex]\[
(-2 \cdot 2) + (1 \cdot 3) + (0 \cdot 4) = -4 + 3 + 0 = -1
\][/tex]
4. Fourth element of the product (position [2,2]):
Take the dot product of the second row of matrix [tex]\(A\)[/tex] with the second column of matrix [tex]\(B\)[/tex]:
[tex]\[
(-2 \cdot -2) + (1 \cdot 5) + (0 \cdot 1) = 4 + 5 + 0 = 9
\][/tex]
Putting all these elements together, we get the product matrix:
[tex]\[
\begin{pmatrix} 15 & 14 \\ -1 & 9 \end{pmatrix}
\][/tex]
Thus, the correct answer is:
[tex]\[
\left[\begin{array}{cc}15 & 14 \\ -1 & 9\end{array}\right]
\][/tex]
