GATE 2016 EE (EE2) – Question 59

For $\dot x=Ax$, vectors α,β are eigenvectors of a 2×2 A with distinct eigenvalues λ1,λ2. For initial state x(0)=α, the response is

Show answer and explanation

Correct answer: (A) $e^{\lambda_1t}\alpha$

Explanation

The autonomous solution is x(t)=eᴬᵗx(0). Since Aα=λ1α, every power satisfies Aⁿα=λ1ⁿα.

Applying the matrix-exponential series therefore gives eᴬᵗα=e^(λ1t)α. The second mode has zero initial coefficient and is not excited. Hence **A**.

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