I was studying the following type of matrices, $$ A = \begin{pmatrix} 1 & x_{12} & \cdots &x_{1n}\\ 0 & x_{22} & \cdots &x_{2n}\\ \vdots\\ 0&\cdots&0&x_{nn} \end{pmatrix},$$ where

- $x_{ij} \in [0,1];$
- $x_{jj} \neq 1$ for all $j > 1;$
- $\displaystyle \sum^n_{i=1}x_{ji} = 1$ for all $j \in \{1,2,\cdots,n\}.$

On random simulation, I am getting
$$
\lim_{N \rightarrow \infty} A^N \rightarrow \begin{pmatrix}
1 & 1 & \cdots &1\\
0 & 0 & \cdots &0\\
\vdots\\
0&\cdots&0&0
\end{pmatrix}.
$$
**My question is**, what is the cause of this behavior?

I tried the following:

Let $A = \mathbf{1} + U$, where $\mathbf{1}$ is the identity matrix of size $n \times n.$

$$
A^N = (\mathbf{1} + U)^N = \sum^N_{k = 0}\binom{N}{k}U^k
$$ but this is certainly not helping much.

**Note that:** A can be understood as the State transition matrix of a Markov Chain, where one can stay in its state or go to a lower state with a probability.

**Additional Observation:** $\displaystyle \lim_{N\rightarrow \infty} U^N = \mathbf{0}$

**Special Thanks:** @tsnao for enhancing clarity and pointing errors.

9more comments