Determinants and multiplicative functionals on quaternion matrices

Determinants of matrices over a field are multiplicative. Does there exist an extension of the definition of determinants of real matrices to quaternion matrices, such that the multiplication theorem holds? This paper proves there does not exist such an extension. We give a universal property on multiplicative functionals from the set of quaternion matrices to the set of quaternion numbers. The theorem tells us, in a sense, the definition of determinants for quaternion matrices is unique.