# Inverse of symmetric matrices

I know that not all symmetric matrices are invertible, but under which conditions are they invertible ? and do all symmetric matrices have to be either positive or negative definite in order to have inverse ??

If you talk about real symmetric matrices then they are invertible iff all eigenvalues are different from zero. (They don't need to be all all positive or all negative). The inverse of $CDC^{-1}$ is simply $C D^{-1} C^{-1}$ where $D$ is diagonal.
The invertibility depends upon the rank, that is the n-by-n matrix is invertible $\iff$ $rank=n$.
Also note that a symmetric matrix is invertible $\iff$ if all eigenvalues are $\neq 0$.