(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Two square invertible matrices, prove product is invertible

1. The problem statement, all variables and given/known data

IfAandBarenxnmatrices of rankn, prove thatABhas rankn.

2. Relevant equations

There is a list in my textbook outlining equivalent statements, such as:

- A is invertible

- rank(A) = n

- nullity(A) = 0

- The column vectors of A are linearly independent.

- and many others...

3. The attempt at a solution

I've been staring at equivalent statements, theorems, and examples, and cannot seem to think of an equivalency once I consider multiplying AB. I guess the main feat of this part would be to prove that, given rank(A) = n and rank(B) = n, AB has, once reduced, neither a row nor column that is zero, which would ultimately lead to rank(AB) = n. However, I'm not sure how to get there.

I've tried aplying the statement of "The reduced row echelon form of A is the indentity matrix," but realized that A would berow equivalentbut notequalto the identity matrix. Anyone have any ideas? Thanks.

EDIT: I realized I can solve this using determinants, but up to the point in the textbook where this proof is requested, they had not been covered, so the question still stands.

2nd edit: I had another realization: Since rank(A) = n and rank(B) = n, then by the Fundamental Theorem of Invertible Matrices, A is a product of elementary matrices and B is a product of elementary matrices. Therefore, AB is a product of elementary matrices and therefor rank(AB) = n. Is this right?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Two square invertible matrices, prove product is invertible

**Physics Forums | Science Articles, Homework Help, Discussion**