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GATE 2018 CE (CE2) – Question 36

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

The matrix $$\begin{bmatrix}2&-4\\4&-2\end{bmatrix}$$ has ______.

  1. real eigenvalues and eigenvectors
  2. real eigenvalues but complex eigenvectors
  3. complex eigenvalues but real eigenvectors
  4. complex eigenvalues and eigenvectors

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Correct answer: (D) complex eigenvalues and eigenvectors

Explanation

Trace is zero and determinant is 12, so the characteristic equation is $\lambda^2+12=0$, giving $\lambda=\pm2\sqrt3\,i$. A nonzero real eigenvector cannot have a non-real eigenvalue for a real matrix. Both eigenvalues and eigenvectors are complex (D).