Is it possible for a matrix A to be equal to the negative of its own transpose?

Question

Is it possible for a matrix A to be equal to the negative of its own transpose? That is, is it possible for ? If so, give an example of a nonzero 2x2 matrix A that satisfies this equation.

Answer 1

Yes it is.  Consider a square matrix A whose elements satisfy:

 

A_ij = -A_ji

 

Note this condition makes all the elements on the diagonal equal to zero.  Then  

 

(A^T)_ij = A_ji

 

But A_ji = -A_ij by the given condition.

 

Therefore

 

(A^T)_ij = -A_ij

 

Then A^T = -A

 

Any square matrix A with the following properties will be equal to its negative transpose:

(i) All diagonal elements are 0

(ii) Every off-diagonal element equals the negative of its corresponding element on the other side of the diagonal

 

Lee gave you a good example of a 2x2 matrix that satisfies this.

Answer 2

Yes. Let A =

 

  0   1

 

 -1   0

 

Then A^T =

 

  0  -1

  1   0

 

= -A

 

Gotta have them zeroes.