The GATE Grind

GATE 2016 CE (CE1) – Question 13

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

If the entries in each column of a square matrix M add up to 1, then an eigenvalue of M is ______.

  1. 4
  2. 3
  3. 2
  4. 1

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) 1

Explanation

Let the row vector of ones be u. Unit column sums imply uA=u, so1 is a left eigenvalue of A. Since A and its transpose have the same characteristic polynomial,1 is also an eigenvalue of A. Thus **1, D**, must occur irrespective of the other eigenvalues.