The State-Transition Matrix of a Constant Coefficient Matrix Is a Matrix Exponential

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b\in\mathbb{R}$ with $a<b$, let $n\in\mathbb{N}$ and $M\in\mathcal{M}_{n}(\mathbb{R})$, and let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be the map with $A(t)=M$ for every $t\in[a,b]$, which is continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v). Let $\Phi_{A}$ be the state-transition matrix of $A$, and for $X\in\mathcal{M}_{n}(\mathbb{R})$ let $e^{X}$ be the exponential of $X$.

(i) $\Phi_{A}(t,s)=e^{(t-s)M}$ for all $s,t\in[a,b]$.