Matrix Invertibility: RREF to Identity

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A matrix A is invertible if and only if its reduced row echelon form (RREF) is the identity matrix. The discussion emphasizes understanding the implications of row operations on matrix algebra, particularly how they relate to the invertibility of A. If the RREF does not yield the identity matrix, it indicates a row of zeros, leading to a determinant of zero and confirming that A is not invertible. The participants suggest that while the proof may seem straightforward, exploring the relationship between row operations and matrix properties can enhance understanding. Overall, the connection between RREF and matrix invertibility is crucial in linear algebra.
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Prove that a matrix A is invertible if and only if its reduced row echelon row is the identity matrix.
 
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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.
 


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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