Prove the following:
a) (A^T)^T=A
b) (A + B)^T= A^T + B^T
c)(AB)^T=(B^T)(A^T)
I'll set you off on the first and then you should see how to do the rest.
Use (i,j) notation: $\displaystyle A = \sum a_{ij}$.
Then by definition: $\displaystyle A^T = \sum a_{ji}$.
So, $\displaystyle (A^T)^T = \left(\sum a_{ji}\right)^T = \sum a_{ij} = A $.
Hope this helps,
pomp.