Project ForgeEngineering Mathematics
Linear Algebra
GATE 2012
Hard
Let A be the 2×2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = −1. Then the eigenvalues of the matrix A¹⁹ are:
- A1024 and −1024
- B1024√2 and −1024√2
- C4√2 and −4√2
- 512√2 and −512√2
Explanation
A = [[1,1],[1,−1]] has trace 0 and determinant (1)(−1)−(1)(1) = −2, giving the characteristic equation λ² − 2 = 0, so the eigenvalues of A are √2 and −√2. If λ is an eigenvalue of A, λ¹⁹ is an eigenvalue of A¹⁹: (√2)¹⁹ = 2^(19/2) = 2⁹·√2 = 512√2, and (−√2)¹⁹ = −512√2 (since 19 is odd). So the eigenvalues of A¹⁹ are 512√2 and −512√2, matching the file's own answer key.
Source: GATE CS 2012 (2012)
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