Prove 3x3 Skew symmetric matrix determinant is equal to zero

In summary, the conversation discusses the proof that the determinant of any odd-ordered matrix is equal to zero. The question is specifically about proving this for a 3x3 matrix using only row and column interchanging. The attempt at a solution involves trying to collect a row of zeros, but it is found that moving a zero into position affects another row or column containing a zero. It is then suggested to evaluate the determinant directly, but the question stipulates not to do so. Another hint is given to use the interchanging of rows and columns, but it is unclear if this is possible. One approach is to show that if |A| = k and -|AT| = k, then k must be zero, and the determinant of
  • #1
Bill333
7
0

Homework Statement


Hi there,

I'm happy with the proof that any odd ordered matrix's determinant is equal to zero. However, I am failing to see how it can be done specifically for a 3x3 matrix using only row and column interchanging.

Homework Equations


I have attached the determinant as an image
Picture1.png

The Attempt at a Solution


So my attempt was to try and collect a row of zeros via interchanging rows/columns. However it seems to be me that trying to move a zero into position by moving its row/column then affects the row/column containing another zero and so I can't ever get them into the same row.
 
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  • #2
Bill333 said:

Homework Statement


Hi there,

I'm happy with the proof that any odd ordered matrix's determinant is equal to zero. However, I am failing to see how it can be done specifically for a 3x3 matrix using only row and column interchanging.

Homework Equations


I have attached the determinant as an image
View attachment 102053

The Attempt at a Solution


So my attempt was to try and collect a row of zeros via interchanging rows/columns. However it seems to be me that trying to move a zero into position by moving its row/column then affects the row/column containing another zero and so I can't ever get them into the same row.

Write it out in detail, using
http://www.coolmath.com/algebra/14-determinants-cramers-rule/02-determinants-3x3-method-1-03
 
  • #3
Why not just evaluate the determinant directly? Since it is specifically a 3 by 3 matrix?

But if you really want to do it using the invariant operations, you need an additional one for this example apart from basic row or column interchanging - the determinant doesn't change when you add a multiple of a column/row to another.
 
  • #4
Thank you for your replies; the question stipulated not to find the determinant directly, and gave a hint concerning the interchanging of rows and columns; I presumed that it did not mean use addition of rows and columns. Fightfish, to clarify, are you saying it is impossible to complete using only row and column swaps?

I attach the question; it is number 9. I was wondering if it could be applied more generally.
IMG_2720.JPG
 
  • #5
Call your matrix of problem 9 (in image) A. If you can apply row or column switches to get from |A| to -|AT, then the determinant is equal to 0.

In other words, if you can do this |A| = -|A1| = |A2| = ... -|AT|, then you're almost done. (Here the A1, A2, and so on are the different matrices you get from the row/column operations.) The reason is that a matrix and its transpose have the same determinant, so if |A| = k, a nonzero number, and -|AT| = k, then k has to be zero.

Now, as far as getting from |A| to |AT|, I have no idea which row/column switcheroos might produce this, but I believe that's the direction they're leading you.
"The details are left to the reader."
 
  • #6
Do you have to use only row and column operations? If not, a simple way to do this would be to recognize that since ##A## is skew-symmetric then ##A = -A^t##, so ##\det(A) = \det(-A^t) = (-1)^3 \det(A^t) = -\det(A^t)##. Now show that the determinant of ##A^t## vanishes, which completes your proof.
 

1. What is a skew symmetric matrix?

A skew symmetric matrix is a square matrix where the entries above the main diagonal are the negative of the corresponding entries below the main diagonal. In other words, the transpose of a skew symmetric matrix is equal to its negative.

2. How do you prove that the determinant of a 3x3 skew symmetric matrix is equal to zero?

To prove that the determinant of a 3x3 skew symmetric matrix is equal to zero, we can use the fact that the determinant of a matrix equals the product of its eigenvalues. Since a skew symmetric matrix always has at least one eigenvalue equal to zero, the determinant must also be equal to zero.

3. Can you provide an example of a 3x3 skew symmetric matrix?

One example of a 3x3 skew symmetric matrix is:

[0 -2 3]

[2 0 -5]

[-3 5 0]

4. Why is it important to know the determinant of a skew symmetric matrix?

The determinant of a skew symmetric matrix is important because it can tell us if the matrix is invertible or not. Since the determinant is equal to zero, a 3x3 skew symmetric matrix is not invertible and therefore cannot be used to solve systems of equations.

5. Can the determinant of a skew symmetric matrix ever be non-zero?

No, the determinant of a skew symmetric matrix can never be non-zero. This is because the determinant is equal to the product of its eigenvalues, and a skew symmetric matrix always has at least one eigenvalue equal to zero.

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