This question involves multiplying two matrices. Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. The given first matrix is 2x2 and the second is 2x2, so their product will also be a 2x2 matrix.

The element in the i-th row and j-th column of the product is computed by multiplying corresponding elements of the i-th row of the first matrix and the j-th column of the second matrix, then summing the results.

For a general product:

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right]$

Final Answer: The third option is the correct product matrix.