Prove that a matrix A is invertible if and only if its reduced row echelon row is theby mathwizarddud Tags: echelon, invertible, matrix, prove, reduced 

#1
Jun3008, 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.




#2
Jun3008, 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. 



#3
Jul108, 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 rowoperations amount to when they rowreduce A to I?
As for the "forward" conjecture, well I can think of something some might find objectionable. If it does not rowreduce 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. 


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