GATE EC 2026 Set 1 Q1 — Control Systems

Consider the differential equation $\dot{\vec{w}}=A \vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$. If $\vec{w}(t)=e^{t} \vec{u}_{x}+e^{-2 t} \vec{u}_{y}$ be the solution to the equation where $\vec{u}_{x}$ and $\vec{u}_{y}$ are unit vectors along the positive $x$ and $y$ axes respectively, then which of the following options is the correct matrix representing $A$?