To find the limiting probability you solve the systems of equations:
- $\vec{\pi}=P\vec{\pi}$
- $\Sigma \pi_j = 1$
and my teacher told us "you could rewrite this as matrices". Having just completed a first intro course in Linear Algebra, I'm curious on how you would do that. Guess it's a matter of simple algebra.
$P$ is already a matrix ($n \times n$). Your first equation is $(P - I) \vec{\pi} = \vec{0}$ where $I$ is the $n \times n$ identity matrix. Your second is $(1,\ldots,1) \vec{\pi} = 1$. So you get the matrix-vector equation $A \vec{\pi} = \vec{b}$ where $A$ is the $(n+1)\times n$ matrix whose first $n$ rows are $P - I$ and last row is all $1$'s, and $\vec{b}$ consists of $n$ $0$'s and then a $1$.
The system is redundant, since the sum of the first $n$ rows of $A$ is $0$ (do you see why?); you can leave out one of those rows to get an $n \times n$ system.