# Skew-symmetric matrices problem ?

1. Jul 19, 2007

### ngkamsengpeter

1. The problem statement, all variables and given/known data

Give an example of two skew-symmetric matrices. Show explicitly that they display the property of skew-symmetry, ie, AB = -BA

2. Relevant equations

3. The attempt at a solution

transpose of (AB) = BA
I just can show that AB=BA but can't show AB=-BA .
Is it (-A)(-B)=AB ?

2. Jul 20, 2007

### Dick

The defining property of a skew-symmetric matrix is transpose(A)=-A. So yes, transpose(AB)=(-B)*(-A)=BA. Beyond that I simply don't follow you. A=[[0,1],[-1,0]]. B=transpose(A)=[[0,-1],[1,0]]. AB=1=BA. AB is not equal to -BA. AB=-BA is property of anticommuting matrices.

Last edited: Jul 20, 2007
3. Jul 20, 2007

### chanvincent

Since AB=BA, AB can't be -BA unless AB = 0

4. Jul 20, 2007

### HallsofIvy

Staff Emeritus
Nonsense. Multiplication of matrices is NOT in general commutative.

5. Jul 20, 2007

### ngkamsengpeter

I think i have misunderstood the questions. My lecturer say the question is we multiply any 2 matrices to get a skew-symmetry matrix AB . Then show that AB=-BA . But i simply can't show it .
I even don't know how to give 2 matrices where the product of these 2 matrices is skew-symmetry matrix .
I know i can use try but i think that is not a good and standard technique .
Anybody has any idea on this question?

6. Jul 20, 2007

### Dick

A=[[-1,-1,0],[0,1,1],[-1,0,1]], B=[[0,1,0],[0,0,1],[1,0,0]].

AB is skew symmetric. AB is not equal to -BA. No wonder you can't show it. Do you mean to add the assumption A and B are symmetric?