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Saturday, 20 August 2011

Math GRE - #19

Let $A$ be a $2\times 2$ matrix for which there is a constant $k$ such that the sum of the entries in each row and each column is $k$. Which of the following must be an eigenvector of $A$?

  1. $\left[\begin{array}{c}
    1\\ 0\end{array}\right]$
  2. $\left[\begin{array}{c}
    0\\ 1\end{array}\right]$
  3. $\left[\begin{array}{c}
    1\\ 1\end{array}\right]$
  • 1 only
  • 2 only
  • 3 only
  • 1 and 2
  • 1, 2, and 3

Solution :

Choice 3 is the only solution.

Since the rows add up to $k$, $\left[\begin{array}{c}1\\ 1\end{array}\right]$ is an eigenvector with eigenvalue $k$. To see that choice 1 and 2 do not work, take the matrix: \[\left[\begin{array}{cc} 1 & 1\\ 1 & 1\end{array}\right]\] which satisfies the given conditions but do not have either choice 1 or choice 2 as eigenvectors.

Bonus problem:
Given a matrix with columns adding up to $k$, prove that $k$ is an eigenvalue (in fact, if the entries in the matrix are all greater than zero, then $k$ is the largest eigenvalue).


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