To find the element in row 1, column 1 of the matrix product [tex]\( A \cdot B \)[/tex], we need to perform matrix multiplication. Here are the steps to calculate the element in row 1, column 1 of the resultant matrix:
Given matrices:
[tex]\[ A = \begin{pmatrix}
1 & 2 & -4 \\
0 & 3 & -1
\end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix}
-1 & 0 \\
2 & 4 \\
7 & 2
\end{pmatrix} \][/tex]
The element in place (1,1) of the product matrix [tex]\( C = A \cdot B \)[/tex] is calculated as follows:
1. Multiply each corresponding pair of elements in the first row of matrix [tex]\( A \)[/tex] with the first column of matrix [tex]\( B \)[/tex].
2. Sum these products.
Mathematically, the calculation is:
[tex]\[ C_{1,1} = (1 \cdot -1) + (2 \cdot 2) + (-4 \cdot 7) \][/tex]
Let's compute step-by-step:
1. Multiply the elements:
- [tex]\( 1 \cdot -1 = -1 \)[/tex]
- [tex]\( 2 \cdot 2 = 4 \)[/tex]
- [tex]\( -4 \cdot 7 = -28 \)[/tex]
2. Sum the products:
[tex]\[ -1 + 4 - 28 = -25 \][/tex]
Thus, the element in row 1, column 1 of the matrix [tex]\( A \cdot B \)[/tex] is:
[tex]\[ \boxed{-25} \][/tex]
