Prove that a matrix A is invertible if and only if its reduced row echelon row is the

by mathwizarddud
Tags: echelon, invertible, matrix, prove, reduced
mathwizarddud is offline
Jun30-08, 09:01 PM
P: 25
Prove that a matrix A is invertible if and only if its reduced row echelon row is the identity matrix.
Phys.Org News Partner Science news on
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
rock.freak667 is offline
Jun30-08, 09:30 PM
HW Helper
P: 6,214
Even though I was never taught linear algebra fully, to do this problem I would consider what would make the matrix A invertible and what would it mean if the RRE form wasn't the identity matrix.

But I am not sure if that would be a valid proof.
Defennder is offline
Jul1-08, 12:08 AM
HW Helper
P: 2,618
This isn't too hard to prove. You can start by asking yourself what a row operation on a matrix translates to in matrix algebra. And what do the matrices corresponding to the row-operations amount to when they row-reduce A to I?

As for the "forward" conjecture, well I can think of something some might find objectionable. If it does not row-reduce A to I, it the RRE form has a row of zeros. That means that the determinant is 0 and hence it is not invertible. I'm sure there's a better way to do this.

Register to reply

Related Discussions
[SOLVED] Two square invertible matrices, prove product is invertible Calculus & Beyond Homework 1
Row-Reduced Echelon Forms Linear & Abstract Algebra 7
Determinant of a matrix using reduced echelon form Precalculus Mathematics Homework 7
reduced row echelon form of matrix Linear & Abstract Algebra 1
reduced row echelon form Introductory Physics Homework 1