Let $I \subset \mathbb{R}$ be a nonempty interval, let $n \in \mathbb{N}$ with $n \geq 1$, let $A: I \to \mathbb{R}^{n \times n}$ be continuous, and let $t_0 \in I$. Then there exists a unique map $\Phi: I \to \mathbb{R}^{n \times n}$ such that $\Phi$ is continuous on $I$, differentiable on the interior of $I$, satisfies $\Phi'(t)=A(t)\Phi(t)$ for every interior point $t$ of $I$, satisfies $\Phi(t_0)=I_n$, where $I_n$ is the $n \times n$ identity matrix, and satisfies $\Phi(t) \in GL(n,\mathbb{R})$ for every $t \in I$. Equivalently, $\Phi$ is the unique principal fundamental matrix for $\dot{x}=A(t)x$ based at $t_0$. Moreover, for every $x_0 \in \mathbb{R}^n$, there exists a unique map $x: I \to \mathbb{R}^n$ that is continuous on $I$, differentiable on the interior of $I$, satisfies $\dot{x}(t)=A(t)x(t)$ for every interior point $t$ of $I$, and satisfies $x(t_0)=x_0$; this solution is given by $x(t)=\Phi(t)x_0$ for every $t \in I$.