Problem16:A square matrix A is called skewsymmetric if, 
> ;
T A = A
Show that

Let's show that the inverse of A is skewsymmetric when A is. Since the operation of taking the transpose commutes with the operation of taking the inverse we have, 
> ;
1 T T 1 (A ) = (A )
since A is skewsymmetric we can write, 
> ;
1 T 1 (A ) = (A)
and now using the obvious fact that for every nonzero scalar t,
inverse(t A) = (1/t) inverse(A) we have, 
> ;
1 T 1 (A ) = A
Thus, the inverse of A is skewsymmetric.
We now show that when A and B are skewsymm. so are the transpose of A, A+B, AB and kA. We prove this by showing that the transpose of each of this matrices is equal to minus the matrix. 
> ;
T T T T (A ) = (A) =  A
this is due to the fact that A is skewsymmetric and the transpose of a number
times a matrix is the number times the transpose of the matrix.
Now for A+B, 
> ;
T T T (A + B) = A + B = A  B = (A + B)
idem for AB, 
> ;
T T T (A  B) = A  B = A + B = (A  B)
finally for kA, with k scalar, 
> ;
T T (kA) = k (A ) = k (A) =  (kA)