I'm not sure what consistency is when applied to matrices, but here's something I knowabout ranks:
The rank of a matrix is equal to the number of linearly independent equations in the system.
A system of equations is "consistent" if there exist at least one solution to the system. If the rank of an n by n coefficient matrix is n, then there exist a unique solution and the system is consistent. If the rank is less than n, then the range of the matrix is a subspace or [itex]R^n[/itex] and there exist a solution if and only if the vector on the right hand side of the equation is in that subspace.