The general solution is \[\mathbf{x}(t)=C_1e^{-t}\begin{pmatrix}1\\2\\\end{pmatrix}+C_2e^{2t}\begin{pmatrix}2\\1\end{pmatrix}\] In a later section, we will discuss the fundamental matrix, and the fundamental matrix for this problem is \[\begin{pmatrix}e^{-t}&2e^{2t}\\2e^{-t}&e^{2t}\end{pmatrix}\]