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GATE 2020 CE (CE2) – Question 37

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

Find eigenvalues of $$P=\begin{bmatrix}0&1&3&0\\-2&3&0&4\\0&0&6&1\\0&0&1&6\end{bmatrix}.$$

  1. 0, 3, 6, 6
  2. 1, 2, 3, 4
  3. 3, 4, 5, 7
  4. 1, 2, 5, 7

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Show answer and explanation

Correct answer: (D) 1, 2, 5, 7

Explanation

Block triangularity makes the characteristic polynomial the product of those of the diagonal 2×2 blocks. They give $(\lambda-1)(\lambda-2)$ and $(\lambda-5)(\lambda-7)$. Thus the eigenvalues are **1, 2, 5, 7, D**.