The GATE Grind

GATE 2022 PI – Question 36

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

When a matrix is squared, which statement is correct?

  1. Both eigenvalues and eigenvectors must change.
  2. Both eigenvalues and eigenvectors are retained.
  3. Eigenvalues are squared and eigenvectors retained.
  4. Eigenvalues are retained but eigenvectors change.

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Correct answer: (C) Eigenvalues are squared and eigenvectors retained.

Explanation

If Av = λv, then $A^2v=A(\lambda v)=\lambda^2v$. Thus each original eigenvector remains an eigenvector and its eigenvalue becomes λ²: **C**. Degenerate squared eigenvalues may additionally enlarge eigenspaces.

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