Let $A$ be a constant $n\times n$ matrix . Consider the system the system \left\{\begin{array}{cc}x'&= Ax\\x(0)&=x_0 \end{array}\right..
$\quad\rm(a)$ Prove that the solution can be expressed in the form $\mathcal{L}_\lambda^{-1}\left\{{\left( {\lambda I-A} \right)^{-1}}\right\}(t)\;x_0$.
Here I can only differentiate and check, but I don't know how to differentiate this. )= Please show me how. And I don't know - what is that lambda? :S
$\quad\rm(b)$ Show that $\mathcal{L}\left\{{e^{At}}\right\}(\lambda)=\left({\lambda I-A} \right)^{-1}.$