I am given that $K=[B\hspace{0.5cm}AB\hspace{0.5cm}A^2B\dots\dots A^{n-1}B]$ is full rank $n$ where $A$ is a column stochastic and $B$ is a positive matrix. Now I define $M=(1-m)A+\frac{m}{n}P$, where $P$ is a matrix having all entries $1$, $m\in(0,1)$. Now could anyone help me to say anthing about rank of this new matrix
$L=[B\hspace{0.5cm}MB\hspace{0.5cm}M^2B\dots\dots M^{n-1}B]$?
$A,B$ matrices are of order $n$, $M$ is also column stochastic.