Termwise <differentiation> of the <matrix exponential> gives $A'=LA$ and $A(0)=I$. Thus $x(t)=A(t)x(0)$ satisfies $x'=Lx$ and the prescribed <initial condition>. Uniqueness for a <linear differential equation> makes it the solution.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 331 / 3 / b / Solution

 Solution (source code)