Project ForgeEngineering Mathematics
Linear Algebra
GATE 2021
Medium
Consider a vector p in 2-dimensional space with direction (counter-clockwise angle with the positive x-axis) θ. Let p be an eigenvector of a 2×2 matrix A with eigenvalue λ, λ>0. Denoting the magnitude of a vector v by ||v||, identify the VALID statement regarding p' = Ap.
- ADirection of p' = λθ, ||p'|| = ||p||
- Direction of p' = θ, ||p'|| = λ||p||
- CDirection of p' = λθ, ||p'|| = λ||p||
- DDirection of p' = θ, ||p'|| = ||p||/λ
Explanation
By definition, if p is an eigenvector of A with eigenvalue λ, then Ap = λp — a pure scalar multiple of p. Since λ>0, p' points in the SAME direction as p (θ unchanged) and has magnitude scaled by λ.
Source: GATE ME 2021 Session 1 (ME-1) (2021)
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