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

In summary, the conversation discusses the problem of proving that the identity matrix cannot be the product of 3 or 5 row exchanges, but can be the product of 2 or 4 exchanges. The process of counting the number of rows that are different from the identity matrix is explained, but the proof is deemed insufficient for not addressing the case of 5 swaps and not offering an explanation for why there can be 0, 3, or 4 different rows after two swaps. The conversation also mentions that the identity matrix cannot be a product of an odd number of row exchanges. A question is posed about whether the row reduction rules for determinants can be taken as given.
  • #1
Ara macao
27
0
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?
 
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  • #2
Is this proof sufficient enough?
No, for two reasons.

(1) You had to prove something about 5 swaps, but have said nothing about that.

(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
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...?
 
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  • #4
Can you take the row reduction rules for determinants as given?
 
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1. What is the identity matrix?

The identity matrix is a square matrix where all elements on the main diagonal are equal to 1 and all other elements are equal to 0. It is denoted by the letter I and is used in linear algebra to represent the identity transformation.

2. How can a matrix be expressed as a product of row exchanges?

A matrix can be expressed as a product of row exchanges by multiplying it with elementary matrices, which are matrices that represent single row operations such as swapping two rows or multiplying a row by a scalar.

3. Why can't the identity matrix be expressed as a product of an odd number of row exchanges?

This is because the determinant of the identity matrix is 1, but the determinant of a matrix obtained by multiplying it with an odd number of row exchanges would be -1. Since the determinant is a multiplicative property, the identity matrix cannot be obtained by multiplying it with an odd number of row exchanges.

4. Can the identity matrix be expressed as a product of an even number of row exchanges?

Yes, the identity matrix can be expressed as a product of an even number of row exchanges. For example, swapping two rows twice would result in the original matrix, and the determinant would remain 1.

5. Is there a proof or mathematical explanation for why the identity matrix cannot be expressed as a product of an odd number of row exchanges?

Yes, this can be proven using the properties of determinants and the fact that the determinant of a matrix obtained by multiplying it with a row exchange matrix is equal to the determinant of the original matrix multiplied by the determinant of the row exchange matrix. This proof can be found in linear algebra textbooks or online resources.

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