# Thread: Rank of Product Of Matrices

1. ## Rank of Product Of Matrices

Question: 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.

Attempt: 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).

2. I would do it like this:
A is a mxn matrix of rank m, now it is obvius that $\displaystyle m\leq n$, why?
So A represents a linear map $\displaystyle f$ with rank $\displaystyle m\leq n$.
B is a nxp matrix of rank n, it is equally obvius that $\displaystyle p\geq n$.
So B represents a linear map $\displaystyle g$ with of rank n.

Now AB represents linear $\displaystyle f\circ g$, or $\displaystyle f(g(\vec v))$
So g has a domain of p dimensions, and the codomain has n dimensions.
f has domain of n dimensions, and the codomain has m dimensions.

So $\displaystyle AB$ represents a map $\displaystyle f\circ g:V\to W$, where V is p dimensional and W is m dimensional. So AB has rank m.

This is what I would do, but there are probably nicer ways. Assuming ofcourse that this is correct.

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