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Circulant matrices over finite fields and over commutative finite chain ringshave been of interest due to their nice algebraic structures and wideapplications. In many cases, such matrices over rings have a closed connectionwith diagonal matrices over their extension rings. In this paper, thedeterminants of diagonal and circulant matrices over commutative finite chainrings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\timesn$ diagonal matrices over ${R}$ of determinant $a$ is determined for allelements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, theenumeration of nonsingular $n\times n$ circulant matrices over ${R}$ ofdeterminant $a$ is given for all units $a$ in $ {R}$ and all positive integers$n$ such that $\gcd(n,q)=1$. In some cases, the number of singular $n\times n$circulant matrices over ${R}$ with a fixed determinant is determined throughthe link between the rings of circulant matrices and diagonal matrices. Asapplications, a brief discussion on the determinant of diagonal and circulantmatrices over commutative finite principal ideal rings is given. Finally, someopen problems and conjectures are posted

Determinants of some Special Matrices over Commutative Finite Chain Rings