## Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1

## Problem 185

Let $A=(a_{ij})$ be an $n \times n$ matrix.

We say that $A=(a_{ij})$ is a **right stochastic matrix** if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have

\[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\]
for $1 \leq i, j \leq n$.

Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.

**(a)**The stochastic matrix $A$ has an eigenvalue $1$.

**(b)** The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.