Theorem.

1. The sum of two symmetric matrices is a symmetric matrix.
2. If we multiply a symmetric matrix by a scalar, the result will be a symmetric matrix.
3. If A and B are symmetric matrices then AB+BA is a symmetric matrix (thus symmetric matrices form a so-called Jordan algebra).
4. Any power Aⁿ of a symmetric matrix A (n is any positive integer) is a symmetric matrix.
5. If A is an invertible symmetric matrix then A^-1 is also symmetric.

Proof. 1 Let A and B be symmetric matrices of the same size. Consider A+B. We need to prove that A+B is symmetric. This means (A+B)^T=A+B. Recall a property of transposes: the transpose of a sum is the sum of transposes. Thus (A+B)^T=A^T+B^T. But A and B are symmetric. Thus A^T=A and B^T=B. So (A+B)^T=A+B and the proof is complete.

Other parts of the theorem are left as an exercise.