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GATE 2024 CE (CE1) – Question 36

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

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?

  1. 1, 2, 5
  2. 1, 3, 4
  3. −5, 1, 2
  4. −5, −1, 2

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Correct answer: (A) 1, 2, 5

Explanation

The trace of the matrix is 8 and its determinant is $2(8 - 1) - 1(2 - 1) + 1(1 - 4) = 14 - 1 - 3 = 10$. The eigenvalues 1, 2 and 5 have the sum 8 and the product 10. The characteristic polynomial is $(\lambda - 1)(\lambda - 2)(\lambda - 5)$ (for example, $\lambda = 1$ gives the singular matrix $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 1 \end{bmatrix}$).