Proposition (Properties of transition matrix)

Given an LTV system and its associated transition matrix $\Phi_{A}(t,t_{0})$ we have the following properties:

1. $$\Phi_{A}(t,t_{0})=\Phi_{A}(t,t_{1})\Phi_{A}(t_{1},t_{0})\quad\forall t_{0},t_{1},t\in J$$ where $J$ is the interval of definition of $A$.
2. $$\Phi_{A}(t,\tau)=\Phi^{-1}(\tau,t)$$