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

Termwise differentiation of the matrix exponential gives $A^{\prime }=LA$ and $A(0)=I$. Thus $x(t)=A(t)x(0)$ satisfies $x^{\prime }=Lx$ and the prescribed initial condition. Uniqueness for a linear differential equation makes it the solution.