# Prove identity matrix cannot be product of an odd number of row exchanges

1. Jul 22, 2006

### Ara macao

Problem: #29 in Strang Linear Algebra

Prove that the identity matrix cannot be the product of 3 row exchanges (or five). It can be the product of 2 exchanges (or 4).

Now, to start, we try to count the number of rows that are different from the identity matrix. For the first row exchange, it's 2. Second, it becomes either 0, 3, or 4. Third, it can go to 1,2,3,4,6. Four, 0,1,2,3,4,5,6,8...

Is this proof sufficient enough?

Last edited: Jul 22, 2006
2. Jul 22, 2006

### Hurkyl

Staff Emeritus
No, for two reasons.

(2) You haven't offered any proof of why, for example, you can have 0, 3, or 4 rows different after two swaps.

(Incidentally, you are wrong. For example, I can easily get 8 rows different after 4 swaps, if my matrix is large enough)

3. Jul 22, 2006

### Ara macao

Oh, sorry, title post was wrong, actually, identity matrix cannot be a product of 3 or 5 row exchanges - but did not say odd number of row exchanges.

hm. After 1 swap, we have 2 rows diferent. After 2 swaps, both rows can swap back to identity, one dif row can swap with one same row, or two same rows swap to different. So we have 0,2,4. And then the thing continues...?

Last edited: Jul 22, 2006
4. Jul 22, 2006

### 0rthodontist

Can you take the row reduction rules for determinants as given?

Last edited: Jul 22, 2006