1. **State the problem:** We need to determine the state transition matrix $\Phi(t)$ (often denoted as $\Phi(t)$ or $cb(t)$) for a given system. 2. **Formula and explanation:** The state transition matrix $\Phi(t)$ for a linear time-invariant system with state matrix $A$ is given by: $$\Phi(t) = e^{At} = \sum_{n=0}^\infty \frac{(At)^n}{n!}$$ This matrix describes how the state evolves over time from an initial state. 3. **Important rules:** - If $A$ is diagonalizable, $A = VDV^{-1}$, then: $$e^{At} = Ve^{Dt}V^{-1}$$ where $e^{Dt}$ is a diagonal matrix with entries $e^{\lambda_i t}$, $\lambda_i$ being eigenvalues of $A$. - If $A$ is not diagonalizable, use Jordan form or series expansion. 4. **Intermediate work:** - Identify matrix $A$ (not provided in the problem, so assume it is given or known). - Compute eigenvalues and eigenvectors of $A$. - Form $V$, $D$, and compute $e^{Dt}$. - Calculate $\Phi(t) = Ve^{Dt}V^{-1}$. 5. **Final answer:** The state transition matrix $cb(t)$ is $\Phi(t) = e^{At}$ computed as above. Since the matrix $A$ is not provided, the explicit form of $cb(t)$ cannot be determined here. Please provide matrix $A$ for a concrete solution.