Let A be an$\displaystyle {m\times n}$ matrix
B be an$\displaystyle {m\times m}$ nonsingular matrix and
C be an$\displaystyle {n\times n}$ nonsingular matrix
Show that rank(A)=rank(BA)=rank(AC)
How you would prove this depends upon the precise definition of "rank" you are using. I am going to use this: If L is a linear transformation from vector space X to vector space Y, the rank(L) is the dimension of L(X).
Let the rank of A be p. Then A maps $\displaystyle R^n$ to a p-dimensional subspace of $\displaystyle R^m$. Since B is non-singular, it maps that p-dimensional subspace of [$\displaystyle R^m$ to a p-dimensional subspace of $\displaystyle R^m$. That is, BA maps $\displaystyle R^n$ to a p-dimensional subspace of $\displaystyle R^m$. Therefore, rank(BA)= p.
Since C is non-singular, it maps $\displaystyle R^n$ to all of $\displaystyle R^n$. A then maps $\displaystyle R^n$ to a p-dimensional subset of $\displaystyle R^m$. That is, AC maps $\displaystyle R^n$ to a p-dimensional subset of $\displaystyle R^m$. Therefore, rank(AB)= p.