(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.)


How do you define the rank of a matrix whose entries are polynomials in $K[X]$?
If you assign some element of $K$ in the entries of such a matrix, what is the rank of the produced matrix (in $M_{mn}(K)$)?  Is it larger, equal, or smaller than that of the original matrix?


EDIT: Here $K$ denotes an arbitrary field, but mostly I'm interested in $\mathbb{R}$ and $\mathbb{F}_p$.

I believe that following expression  is unique solution to the riddle :

$2^6-63=1$
