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GATE 2021 ME (ME2) – Question 11

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 1 mark · Multiple choice

Consider an $n\times n$ matrix $A$ and a non-zero $n\times1$ vector $p$. Their product $Ap=\alpha^2p$, where $\alpha\in\mathbb R$ and $\alpha\notin\{-1,0,1\}$. Based on the given information, the eigen value of $A^2$ is:

  1. $\alpha$
  2. $\alpha^2$
  3. $\sqrt\alpha$
  4. $\alpha^4$

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Correct answer: (D) $\alpha^4$

Explanation

Apply A once more: $A^2p=A(\alpha^2p)=\alpha^2Ap=\alpha^4p$. Since p is nonzero, α⁴ is an eigenvalue of A².

**Answer: D**. Squaring a matrix squares the corresponding eigenvalue, not the given symbol alone.