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

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

Other parts of the theorem are left as an exercise.