The GATE Grind

GATE 2026 EC – Question 11

Engineering Mathematics · Differential Equations · 1 mark · Multiple choice

Consider the differential equation $\dot{\vec{w}} = A\vec{w}$, with $\vec{w}(t=0) = \begin{bmatrix}1\\1\end{bmatrix}$. If $\vec{w}(t) = e^{t}\vec{u}_x + e^{-2t}\vec{u}_y$ 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?

  1. $\begin{bmatrix}1&0\\0&-2\end{bmatrix}$
  2. $\begin{bmatrix}-1&0\\0&2\end{bmatrix}$
  3. $\begin{bmatrix}0&-2\\1&0\end{bmatrix}$
  4. $\begin{bmatrix}0&2\\-1&0\end{bmatrix}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\begin{bmatrix}1&0\\0&-2\end{bmatrix}$

Explanation

$\vec w=[e^t,\,e^{-2t}]^T$ gives $\dot{\vec w}=[e^t,\,-2e^{-2t}]^T=\mathrm{diag}(1,-2)\vec w$. So $A=\mathrm{diag}(1,-2)$.