1. ## Rank and nonsingularity

Let A be an ${m\times n}$ matrix
B be an ${m\times m}$ nonsingular matrix and
C be an ${n\times n}$ nonsingular matrix

Show that rank(A)=rank(BA)=rank(AC)

2. Originally Posted by altave86
Let A be an ${m\times n}$ matrix
B be an ${m\times m}$ nonsingular matrix and
C be an ${n\times n}$ 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 $R^n$ to a p-dimensional subspace of $R^m$. Since B is non-singular, it maps that p-dimensional subspace of [ $R^m$ to a p-dimensional subspace of $R^m$. That is, BA maps $R^n$ to a p-dimensional subspace of $R^m$. Therefore, rank(BA)= p.

Since C is non-singular, it maps $R^n$ to all of $R^n$. A then maps $R^n$ to a p-dimensional subset of $R^m$. That is, AC maps $R^n$ to a p-dimensional subset of $R^m$. Therefore, rank(AB)= p.