Here is the 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.