A Functional Equation Related to Determinant of Some 3 X3 Symmetric Matrices and Its Pexiderized Form

In this work, we present the general solution of a functional equation f (ux + vy,uyn+ zw)= f (x,y,z)+ f (u,v,w)+ f  (x,y,z) f (u,v,w) for all x,y,u,v,w, z   , which arises from determinant of some symmetric 3 x 3 matrices. We also determine the general solution of its Pexiderized version f (ux+vy,uy + vx,zw)= (x,y,z)+ h (u,v,w)+  (x,y,z) n (u,v,w) for all x,y,u,v,w,z  ,without any regularity assumptions on unknown functions f,,h,,n :³   .

Keywords: Determinant of matrix, functional equation, logarithmic function, multiplicative function

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E-mail: charinthip@mathstat.sci.tu.ac.th