To find the matrix product [tex]\( A \times B \)[/tex], we perform the matrix multiplication of [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Given the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A=\begin{pmatrix}
5 & 7 & 2 \\
4 & -1 & 3 \\
6 & 8 & -5
\end{pmatrix}
\][/tex]
[tex]\[
B=\begin{pmatrix}
6 & 11 & -4 \\
2 & 1 & -5 \\
3 & -9 & 6
\end{pmatrix}
\][/tex]
To find the element in the [tex]\(i\)[/tex]-th row and [tex]\(j\)[/tex]-th column of the product matrix [tex]\(C = AB\)[/tex], we calculate 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].
The matrix multiplication result [tex]\(AB\)[/tex] is:
[tex]\[
AB=\begin{pmatrix}
50 & 44 & -43 \\
31 & 16 & 7 \\
37 & 119 & -94
\end{pmatrix}
\][/tex]
So, matching each value to the correct entry in matrix [tex]\(AB\)[/tex]:
- [tex]\( AB_{11} = 50 \)[/tex]
- [tex]\( AB_{12} = 44 \)[/tex]
- [tex]\( AB_{13} = -43 \)[/tex]
- [tex]\( AB_{21} = 31 \)[/tex]
- [tex]\( AB_{22} = 16 \)[/tex]
- [tex]\( AB_{23} = 7 \)[/tex]
- [tex]\( AB_{31} = 37 \)[/tex]
- [tex]\( AB_{32} = 119 \)[/tex]
- [tex]\( AB_{33} = -94 \)[/tex]
Thus, the specific values are assigned to the corresponding entries as shown above.
