# Linear AlgebraWatch

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#1
If is an square matrix in reduced row echelon form and contains no zero rows why must it be the identity square matrix ?

Would appreciate any help thanks.

EDIT: I can intuitively see it but am struggling to find a formal proof- all the Linear Algebra books where I have seen this neglect to give any kind of extra mention let alone give a proof- it's not that trivial to completely skip is it?
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4 years ago
#2
This isn't very rigorous, but perhaps intuitively gives you the answer (and will hopefully help you realise why it's true).

What makes a matrix R be in RRE form?

1) Zero rows of R are below non-zero rows
2) In the non-zero rows, the leading entry is 1
3) The leading coefficient of a row is always strictly to the 'right' of the leading entry of the row above it
4) If a column contains a leading entry of some row then all other entries in the column are zero

Now, the important point here is that there are no zero rows, this completely restricts where the leading coefficients can be. Imagine the leading entry of the first row is not in the first position (i.e not in the (i,j) = (1,1) position) and instead in position k > 1, then the leading entry in the second row has to appear after k+1, third row has to appear after k+2 etc. by property 3) of RRE form. In an n x n matrix, what you're going to end up with is zero rows (can you see that?). This forces the leading coefficient of the first row to be in the (1,1) position. You can continue this argument inductively on the second/third rows to see that what you must have (just from properties 2 and 3) is an upper triangular matrix with 1s on the main diagonal.

You can then use another property of RRE form to deduce that, in fact, all entries not on the main diagonal have to be zero.

(sorry, I've probably given too much away, but you should form this argument much more quantitatively in an answer)
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#3
(Original post by Noble.)
...
Thanks very much- much clearer after a little experimenting with 2x2 to 4x4s- will try to do in rigorous way later if I am not lazy
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