To find the element in the second row and first column (element \(2,1\)) of the product of the two given matrices, we'll follow the matrix multiplication process. This involves taking the dot product of the second row of the first matrix and the first column of the second matrix.
The given matrices are:
[tex]\[ A = \left[\begin{array}{rrr}
2 & 0 & -8 \\
-1 & 10 & -9
\end{array}\right] \][/tex]
and
[tex]\[ B = \left[\begin{array}{rrr}
-2 & -7 & 3 \\
-5 & 4 & -8 \\
2 & -4 & -6
\end{array}\right] \][/tex]
We need the element in the second row and first column of the product matrix \( AB \). This means we need the dot product of the second row of \( A \) and the first column of \( B \).
Second row of \(A\):
[tex]\[ [-1, 10, -9] \][/tex]
First column of \( B \):
[tex]\[ [-2, -5, 2] \][/tex]
Now, calculate the dot product:
[tex]\[
(-1) \cdot (-2) + (10) \cdot (-5) + (-9) \cdot (2)
\][/tex]
This translates to:
[tex]\[
( -1 ) \cdot ( -2 ) + ( 10 ) \cdot ( -5 ) + ( -9 ) \cdot ( 2 )
\][/tex]
Perform each multiplication:
[tex]\[
( 2 ) + ( -50 ) + ( -18 )
\][/tex]
Sum the results:
[tex]\[
2 - 50 - 18 = -66
\][/tex]
Thus, the element in position \(2,1\) of the product matrix is:
[tex]\[ \boxed{-66} \][/tex]
