1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Singularity/Invertibility of Matrix Product

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that A and B are square matrices of the same order. Prove that if AB is invertible, then A and B are both invertible.


    2. Relevant equations



    3. The attempt at a solution
    I attempted to prove the contrapositive, i.e. if at least one of A,B is singular, then AB is singular. I proved that if B is singular, then AB is singular, but I have not been able to prove that if A is singular, then AB is singular.

    I know that this somehow involves the notion that singular matrices such as C have non-trivial solutions to the homogenous system Cx = 0. But I can't apply it correctly because I don't know the direction to follow.

    Any hints?

    BiP
     
  2. jcsd
  3. Dec 3, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If A is singular then Av=0 for some nonzero vector v. Since you've already shown B is nonsingular then v=Bu for some nonzero vector u, yes?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Singularity/Invertibility of Matrix Product
  1. Invertible matrix (Replies: 14)

  2. Invertible matrix help (Replies: 1)

Loading...