GATE 2012 CS (CS1) – Question 21

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

A is 2×2 with a₁₁=a₁₂=a₂₁=1 and a₂₂=−1. Find the eigenvalues of A¹⁹.

1024 and −1024
$1024\sqrt2,-1024\sqrt2$
$4\sqrt2,-4\sqrt2$
$512\sqrt2,-512\sqrt2$

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Correct answer: (D) $512\sqrt2,-512\sqrt2$

Explanation

A²=2I, so A¹⁹=2⁹A. Eigenvalues of A are ±√2, hence those of A¹⁹ are ±512√2. This agrees with raising each original eigenvalue to the nineteenth power. Thus D.

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