1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $\mathbf{\Phi}(t)$) for a given system. 2. **Formula and explanation:** The state transition matrix $\Phi(t)$ for a linear time-invariant system with system matrix $A$ is given by: $$\Phi(t) = e^{At} = \sum_{k=0}^\infty \frac{(At)^k}{k!}$$ This matrix describes how the state evolves over time from an initial state. 3. **Important rules:** - If $A$ is diagonalizable, $A = PDP^{-1}$, then: $$\Phi(t) = Pe^{Dt}P^{-1}$$ where $D$ is diagonal and $e^{Dt}$ is the diagonal matrix with entries $e^{\lambda_i t}$. - If $A$ is not diagonalizable, use Jordan form or other methods. 4. **Intermediate work:** Since the problem does not provide $A$, the general form is: $$\Phi(t) = e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots$$ where $I$ is the identity matrix. 5. **Explanation:** The state transition matrix is the matrix exponential of $At$. It propagates the state vector from time 0 to time $t$. **Final answer:** $$\boxed{\Phi(t) = e^{At} = \sum_{k=0}^\infty \frac{(At)^k}{k!}}$$