## Pages

News: Currently the LaTeX and hidden solutions on this blog do not work on Google Reader.
Email me if you have suggestions on how to improve this blog!

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

This webpage is LaTeX enabled. To type in-line formulae, type your stuff between two '\$'. To type centred formulae, type '$' at the beginning of your formula and '$' at the end.