Systems of Differential Equations · Challenge problem

The vector function $\mathbf x(t)=\begin{pmatrix}e^{t}\cos t\\ e^{t}(\cos t+\sin t)\end{pmatrix}$ is a solution of $\mathbf x'=A\mathbf x$ for a certain constant real $2\times2$ matrix $A$. Find the sum of the four entries of $A$.