Existence and Uniqueness Theorem for 1st Order IVPs

Theorem
If the $n \times n$ matrix function $A(t)$ and vector function $b(t)$ are continuous on an interval $I$, then the Initial Value Problem
 * $ x' = A(t)x + b(t), \, x(t_0) = x_0 $

(where $t_0 \in I$) has a unique solution that exists on all of $I$.