The GATE Grind

GATE 2017 EC – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider the $5 \times 5$ matrix

$$A = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 2 & 3 & 4 \\ 4 & 5 & 1 & 2 & 3 \\ 3 & 4 & 5 & 1 & 2 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}.$$

It is given that $A$ has only one real eigenvalue. Then the real eigenvalue of $A$ is

  1. $-2.5$
  2. 0
  3. 15
  4. 25

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Correct answer: (C) 15

Explanation

Every row of the matrix adds up to $1 + 2 + 3 + 4 + 5 = 15$. So multiplying $A$ by the vector of all ones gives 15 times that vector, which makes 15 an eigenvalue. Since the matrix has only one real eigenvalue, it is 15.