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. $\left[\begin{array}{c}
   1\\ 1\end{array}\right]$

Choices:

• 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).