On two functional equations and their solutions

The present work aims to determine the solution f : R2 → R of the equation f(ux − vy, uy − vx) = f(x, y) + f(u, v) + f(x, y) f(u, v) for all x, y, u, v ∈ R without any regularity assumption. The solution of the functional equation f(ux+ vy, uy− vx) = f(x, y) + f(u, v) + f(x, y) f(u, v) is also determined. The methods of solution of these equations are simple and elementary. These two equations arise in connection with the characterizations of determinant and permanant of two-by-two symmetric matrices, respectively.