GATE EC 2026 Set 1 — Question 11
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Control Systems → Differential Equations → Eigenvalues & Eigenvectors
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Question
Consider the differential equation , with .If be the solution to the equation where and are unit vectors along the positive x and y axes respectively, then which of the following option correct matrix representing ?
Correct answer
(A) bmatrix 1 & 0 \ 0 & -2 bmatrix
Solution
Given the solution vector . We can write this as a column vector:
First, we find the derivative of with respect to :
The given differential equation is . Substituting the expressions for and :
Let . Then:
Comparing the components:
1)
2) For these equations to hold for all , we must equate the coefficients of and on both sides.
From equation (1):
Coefficient of :
Coefficient of : From equation (2):
Coefficient of :
Coefficient of : Therefore, the matrix is:
We can also verify the initial condition :
, which matches the given condition.This corresponds to option (A).
First, we find the derivative of with respect to :
The given differential equation is . Substituting the expressions for and :
Let . Then:
Comparing the components:
1)
2) For these equations to hold for all , we must equate the coefficients of and on both sides.
From equation (1):
Coefficient of :
Coefficient of : From equation (2):
Coefficient of :
Coefficient of : Therefore, the matrix is:
We can also verify the initial condition :
, which matches the given condition.This corresponds to option (A).
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