Let K be a field with 1 1 notequalto 0 Show that every matr

Let K be a field with 1 + 1 notequalto 0. Show that every matrix A elementof K^n, n can be written as A = M + S with a symmetric matrix M elementof K^n, n (i.e., M^T = M) and a skew-symmetric matrix S elementof K^n, n (i.e., S^T = -S). Does this also hold in a field with 1 + 1 =0? Give a proof or a counterexample.

Solution

Let us chose M = (A+A^T)/2 and S = (A-A^T)/2

Now M^T = (A+A^T)^T/2 = (A^T + (A^T)^T)/2 = (A^T + A)/2 = (A + A^T)/2 = M

=> M^T = M

Hence M is a symmteric matrix

Now consider S^T = (A - A^T)^T/2 = (A^T - (A^T)^T)/2 = (A^T - A)/2 = - (A - A^T)/2 = -S

=> S^T = -S

Hence S is a skew symmetric matrix

So we get that A = (A+A^T)/2 + (A-A^T)/2 = M + S

If 1+1=0 in K

Then suppose A is matrix with all the entries 1

Then A+A^T = 0 matrix

And A-A^T = 0 matrix

Hence A = 0 matrix which is not true

So the result doen\'t hold when 1+1=0 in a field