In 2014, Gupta and Ray proved that the circulant involutory matrices over the finite field F2m can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order 2d × 2d over finite fields of characteristic 2. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in 2022, Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this paper establishes a link between the trace of associated diagonal matrices and the MDS property of matrices over the finite field F2m. Given that existing constructions of circulant MDS matrices rely on exhaustive search methods, our result introduces a necessary condition for a circulant semi-orthogonal (or semi-involutory) matrix to be MDS. Specifically, we prove that for circulant semi-orthogonal matrices of even order and circulant semi-involutory matrices, if the trace of the associated diagonal matrices is non-zero, the matrix cannot be MDS.