What Is the Rank of the Product of Two Matrices?

[JG89]

Homework Statement

Let A be an m * n matrix with rank m and B be an n * p matrix with rank n. Determine the rank of AB. Justify your answer.

Homework Equations

The Attempt at a Solution

I don't really know where to start off, but I have some things that might help me. I know that the rank of a matrix is equal to the number of linearly independent rows in it, and I also know that if A and B are two matrices, then rank(AB) <= rank(A) and also rank(AB) <= rank(B).

I'm sure if I was given a push in the right direction I could solve this.

 

[Dick]

What's the relation between the rank of a matrix and the dimension of the image?