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GATE 2021 ME (ME1) – Question 36

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 2 marks · Multiple choice

Consider a vector $p$ in 2-dimensional space. Let its direction (counter-clockwise angle with the positive x-axis) be $\theta$. Let $p$ be an eigenvector of a $2\times2$ matrix $A$ with corresponding eigenvalue $\lambda$, $\lambda>0$. If we denote the magnitude of a vector $v$ by $\|v\|$, identify the VALID statement regarding $p^{\prime}$, where $p^{\prime}=Ap$.

  1. Direction λθ, magnitude $\|p\|$
  2. Direction θ, magnitude $\lambda\|p\|$
  3. Direction λθ, magnitude $\lambda\|p\|$
  4. Direction θ, magnitude $\|p\|/\lambda$

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Correct answer: (B) Direction θ, magnitude $\lambda\|p\|$

Explanation

By the eigenvector definition, p′=λp. Positive scalar multiplication preserves direction and multiplies length by λ.

Therefore direction remains θ and $\|p^{\prime}\|=\lambda\|p\|$, **B**. A negative eigenvalue would reverse direction, but is excluded.