Using generalized form of the Peano-Baker series, the author gives a closed form of the transition matrix for the linear system

${x}^{{\Delta}}\left(t\right)=A\left(t\right)x\left(t\right)$ on an arbitrary time scale

$\mathbb{T}$ (i.e., some nonempty closed subset of the real axis

$\mathbb{R}$), where

$A\left(t\right)$ is an

$n\times n$ matrix function,

${x}^{{\Delta}}\left(t\right)$ is so-called delta derivative. When

$A\left(t\right)\equiv A$ 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

$A$ with infinitely delta differentiable functions as coefficients.