Question

B is a square matrix., and A = B+B^T. Prove that every such A is symmetric.

Answer 1

must be symmetric. If B has entries (bij) and B^T has entries (cij) then since it is the transpose, cij = bji. Therefore in the sum A = B+B^T each aij = bij + cij = bij +bji. Symmetric means aij = aji:
aij = bij + bji = bji + bij = aji.

Answer 2

hope this is clear. if not let me know.