## Linear Algebra: A Modern Introduction

$A^2=\begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix}^2=\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}=I$ Thus $A^3=A^2A=IA=A$ Hence if $n$ is even $A^n=I$, if $n$ is odd $A^n=A$. Thus no power of $A$ has only positive entries, so it cannot be regular.