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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
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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.
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Jun30-08, 09:30 PM
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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.
Jul1-08, 12:08 AM
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P: 2,616
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.

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