Using generalized form of the Peano-Baker series, the author gives a closed form of the transition matrix for the linear system
on an arbitrary time scale
(i.e., some nonempty closed subset of the real axis
is so-called delta derivative. When
is a constant matrix then the transition matrix has the form of a generalized matrix an exponential on the time scale. Using the generalized Laplace transform, an explicit representation of the time scale matrix exponential is obtained. It is shown that the matrix exponential can be expressed also as a finite sum of powers of the matrix
with infinitely delta differentiable functions as coefficients.