To multiply two matrices, we follow the rules of matrix multiplication. For two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], where [tex]\( A \)[/tex] is of size [tex]\( m \times n \)[/tex] and [tex]\( B \)[/tex] is of size [tex]\( n \times p \)[/tex], the resulting matrix [tex]\( C \)[/tex] will be of size [tex]\( m \times p \)[/tex]. Each element [tex]\( c_{ij} \)[/tex] in [tex]\( C \)[/tex] is obtained by taking 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].
Given the matrices:
[tex]\[ A = \left(\begin{array}{cc}
4 & 2 \\
3 & 6
\end{array}\right) \][/tex]
[tex]\[ B = \left(\begin{array}{cc}
8 & 1 \\
2 & 5
\end{array}\right) \][/tex]
We will now find the resulting matrix [tex]\( C \)[/tex] by calculating each element step-by-step:
1. Calculate [tex]\( c_{11} \)[/tex]:
[tex]\[
c_{11} = 4 \cdot 8 + 2 \cdot 2 = 32 + 4 = 36
\][/tex]
2. Calculate [tex]\( c_{12} \)[/tex]:
[tex]\[
c_{12} = 4 \cdot 1 + 2 \cdot 5 = 4 + 10 = 14
\][/tex]
3. Calculate [tex]\( c_{21} \)[/tex]:
[tex]\[
c_{21} = 3 \cdot 8 + 6 \cdot 2 = 24 + 12 = 36
\][/tex]
4. Calculate [tex]\( c_{22} \)[/tex]:
[tex]\[
c_{22} = 3 \cdot 1 + 6 \cdot 5 = 3 + 30 = 33
\][/tex]
Putting these values together, the resulting matrix [tex]\( C \)[/tex] is:
[tex]\[ C = \left(\begin{array}{cc}
36 & 14 \\
36 & 33
\end{array}\right) \][/tex]
Therefore, the product of the given matrices is:
[tex]\[ \left(\begin{array}{cc}
36 & 14 \\
36 & 33
\end{array}\right) \][/tex]
