Diagonalization & Matrix Powers · Superchallenge problem

Let $K = \begin{pmatrix}2 & -1 & 0\\-1 & 2 & -1\\0 & -1 & 2\end{pmatrix}$ and let $\mathbf{x}(t)$ solve $\mathbf{x}' = -K\mathbf{x}$ with $\mathbf{x}(0) = (1, 0, 0)$. Evaluate $\displaystyle\int_0^\infty \lVert\mathbf{x}(t)\rVert^2\,dt$.