  In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance
Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher,
drawing significant attention to circulant MDS matrices. This block cipher is
now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim
introduced cyclic matrices by modifying the permutation of circulant matrices
and established the existence of MDS property for orthogonal left-circulant
matrices, a notable subclass within cyclic matrices. While circulant matrices
have been well-studied in the literature, the properties of cyclic matrices are
not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a
subclass of cyclic matrices. In this article, we first establish a permutation
equivalence between a cyclic matrix and a circulant matrix. We explore
properties of cyclic matrices similar to $g$-circulant matrices. Additionally,
we determine the determinant of $g$-circulant matrices of order $2^d \times
2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a
finite field of characteristic $2$. Furthermore, we prove that this result
holds for any cyclic matrix.
