1
$\begingroup$

The author of the book I'm reading says that the following proof has a gap,

but I don't know where it is.

Here is the proof which the author claims that it has a gap.

First, The author defines that 'row-equivalentness',

$A$ and $B$ are row-equivalent if there exist elementary matrices $E_i$ such that

$$E_1 E_2 \cdots E_k A = B.$$

where elementary matrix is the matrix corresponding to the elementary row operations. (exchange rows, multiply a row by given scalar, add $i$th row by $c$ multiple of $j$th row)

And shows that row-equivalentness is an equivalence relation.

So if $A$ had $R_1, R_2$ as its row reduced echelon form,

then $R_1 \equiv R_2$.

But the only way in which apply elementary row operations to row reduced echelon form to make

again row reduced echelon form is to exchange zero rows.

Therefore $R_1 = R_2$, i.e., its row reduced echelon form is unique.

Where is the gap? I can't see where it is.

$\endgroup$
1
  • $\begingroup$ Multiplying a zero row by a constant, or adding a multiple of a zero row to another row will also leave the RREF unchanged. Not sure whether that's what the author means. $\endgroup$
    – David
    Mar 21, 2023 at 2:44

1 Answer 1

0
$\begingroup$

To my mind, there’s a gap in the claim that the only way to apply elementary row operations to a reduced row echelon form so as to obtain a reduced row echelon form again is to exchange zero rows.

As David pointed out in a comment, that’s not even true for individual elementary row operations, since you can also multiply a zero row by a constant or add a multiple of it to another row. But that’s not a substantial problem for the proof, since those operations also leave the matrix unchanged, so you could replace that claim by the claim that applying elementary row operations to a reduced row echelon form so as to obtain a reduced row echelon form necessarily leads to the same matrix.

This improved claim is true for individual elementary row operations. The gap is that it’s not obvious that it’s also true for any product of elementary row operations.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .