A note on the matrix solution of the problem \(X^{\Delta} = AX, X(t_{0}) = I\) (Q1779581)

Let \({\mathbb T}\) be a time scale, i.e. a nonempty closed subset of \({\mathbb R}\). The author considers special time scales for which the set of certain ``pathological'' points (which are left-dense and right-dense at the same time) has the Lebesgue measure zero. For such time scales, the author derives a representation of the fundamental matrix \(X(t)\) of the linear autonomous dynamic system \(X^\Delta=AX\), where \(A\) is an \(n\times n\) regressive matrix. This is achieved by considering certain (infinite) matrix product with the usual matrix exponential.