Diagonalization & Matrix Powers · Challenge problem

Two coupled chemical concentrations evolve by $\mathbf x' = A\mathbf x$ with $A = \begin{pmatrix}-3/2 & 1/2\\1/2 & -3/2\end{pmatrix}$ and $\mathbf x(0) = \begin{pmatrix}5\\1\end{pmatrix}$. A toxicologist needs the total cumulative exposure to the first chemical, $\displaystyle\int_0^{\infty} x_1(t)\,dt$. Diagonalize $A$ and find it.