Can anyone explain this conceptual problem?

If an $n \times n$ matrix is in row reduced echelon form, explain why it is either the identity matrix or else has a row of zeroes?


  • $\begingroup$ The choice of a good answer here depends a lot on what you already know; there are a lot of options, but some require prior knowledge that you might not have. Do you already know that a linear system $Ax=b$ has either 0,1, or infinitely many solutions? Each of these cases can be "read off" from the echelon form (or reduced echelon form) of the augmented matrix for the system. $\endgroup$ – Ian May 17 '15 at 19:52
  • $\begingroup$ Do you have an example of a matrix that is in reduced row echelon form but isn't the identity matrix? Why can't you get it in the form of the identity matrix? $\endgroup$ – Maths student May 17 '15 at 19:52

If it doesn't have row of zeros then all the rows will have a leading 1.Which will make it a identity matrix.

  • $\begingroup$ @pasie15 The rule is for RREF (row reduced echelon matrix) not REF (reduced echelon matrix) $\endgroup$ – Shubham Ugare May 17 '15 at 19:54
  • $\begingroup$ In the texts I've seen (Lay and Strang), the form given by "rref" is called "reduced echelon form", not "row reduced echelon form". $\endgroup$ – Ian May 17 '15 at 19:55
  • $\begingroup$ A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions: It is in row echelon form. Every leading coefficient is 1 and is the only nonzero entry in its column.(wiki) $\endgroup$ – Shubham Ugare May 17 '15 at 19:55
  • $\begingroup$ okay..It was different in the book i read..but the answer is according to ref itself.. $\endgroup$ – Shubham Ugare May 17 '15 at 19:56

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