How to prove this matrix is a irreducible matrix?

Question

Suppose A is a $n\times n$ square matrix$$A= \begin{bmatrix} 0 &1 &0 &\cdots &0\\ 0 &0 &1&\cdots &0\\ \vdots &\vdots&\vdots & &\vdots\\ 0&0&0&\cdots&1\\ 1 &0&0&\cdots&0 \end{bmatrix} $$ How to prove A is a irreducible a matrix.

A is irreducible if and if only $(I+A)^{n-1}$ is a postive matrix which means that all the entris is positive. Note that $(I+A)^{n-1}=\sum_{k=0}^{n-1} \binom {n-1} {k}A^k$ and A is nonnegative, thus $A$ is irreducible if and only if $\sum_{k=0}^{n-1}A^k$ is a positive matrix.

Answer 1

Suppose $e_1 = (1,0,\cdots,0),e_2 = (0,1,0,\cdots,0),\cdots,e_n=(0,0,\cdots,0,1)$. Let $e_0= e_n$.

We have $$Ae_k=e_{k-1} ,k=1,2,\cdots,n$$

We have $(\sum_{k=0}^{n-1}A^k)e_i = (e_1,e_2,\cdots,e_n)(1,1,\cdots,1)^T$ for $i=1,2,\cdots,n$.

Then we can conclude that$$\sum_{k=0}^{n-1}A^k=(\sum_{k=0}^{n-1}A^k)I=(\sum_{k=0}^{n-1}A^k)(e_1,e_2,\cdots,e_n) = \begin{bmatrix} 1&1&\cdots &1\\ 1&1&\cdots &1\\ \vdots &\vdots& & 1\\ 1& 1&\cdots &1 \end{bmatrix} $$

which is a positive matrix.

Thus the matrix $A$ is a irreducible matrix.