GATE ECE 2026 | Question: 1

Question

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 $A$?

• A. $\left[\begin{array}{cc}1 & 0 \\ 0 & -2\end{array}\right]$
• B. $\left[\begin{array}{cc}-1 & 0 \\ 0 & 2\end{array}\right]$
• C. $\left[\begin{array}{cc}0 & -2 \\ 1 & 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0 & 2 \\ -1 & 0\end{array}\right]$

Comments

 

Given wbar(t)=e^t ux​+e^−2t uy , here 1 and -2 are ploes of transfer fuction 
In state space,
                   Poles of transfer function = Eigen values of State Matrix A
Hence it is Option A