The following identity is proved: \[ \det (X+A)=\sum_{\lambda \subseteq | n|}\sum_{\sigma \in D(\lambda)}(-1)^{| \lambda | +\iota (\sigma)}X_{\lambda c}per(A_{\sigma}), \] where A is an \(n\times n\) matrix, X is a diagonal matrix, D(\(\lambda)\) is the set of ordered partitions of \(\lambda\), \(\iota\) (\(\sigma)\) is the number of nonempty parts of \(\sigma\). The recurrent relation by \textit{T. Muir} [A relation between permanents of determinants. Proc. Roy. Soc. Edinburgh 22, 134-136 (1897)] follows from the above identity. Expansions for \(per(X+A)\), \(per^{-1}(I-XA)\), \(\det^{-1}(I-XA)\) are also demonstrated, where I denotes the identity matrix, \(\bar m=(m_ 1,m_ 2,...,m_ n)\) the nonnegative integer vector of n dimensions with coordinate sum \(| \bar m|\).