The correct answer is $\boxed{\left[ {\begin{array}{*{20}{c}} {38}&{28} \ {32}&{56} \end{array}} \right]}$.
To find the product of two matrices, we can use the following formula:
$$(A \times B){ij} = \sum{k=1}^n A_{ik} B_{kj}$$
where $A$ is an $m \times n$ matrix, $B$ is an $n \times p$ matrix, and $(A \times B)_{ij}$ is the element in the $i$th row and $j$th column of the product matrix.
In this case, $A$ is a 2 $\times$ 2 matrix and $B$ is a 2 $\times$ 2 matrix. Therefore, the product matrix will be a 2 $\times$ 2 matrix.
To find the element in the first row and first column of the product matrix, we multiply the first row of $A$ with the first column of $B$ and add the products together. This gives us:
$$(A \times B)_{11} = 1 \cdot 3 + 5 \cdot 8 = 38$$
To find the element in the second row and first column of the product matrix, we multiply the second row of $A$ with the first column of $B$ and add the products together. This gives us:
$$(A \times B)_{21} = 6 \cdot 3 + 2 \cdot 8 = 32$$
To find the element in the first row and second column of the product matrix, we multiply the first row of $A$ with the second column of $B$ and add the products together. This gives us:
$$(A \times B)_{12} = 1 \cdot 7 + 5 \cdot 4 = 28$$
To find the element in the second row and second column of the product matrix, we multiply the second row of $A$ with the second column of $B$ and add the products together. This gives us:
$$(A \times B)_{22} = 6 \cdot 7 + 2 \cdot 4 = 56$$
Therefore, the product matrix is:
$$(A \times B) = \left[ {\begin{array}{*{20}{c}} {38}&{28} \ {32}&{56} \end{array}} \right]$$