Determining if a symmetric matrix is neither positive nor negative semi definite

Given any matrix A, how can I determine whether the following system is neither positive semi definite nor negative semi definite?

\begin{pmatrix} I & A \\ A^T & 0 \\ \end{pmatrix}

If $A$ is invertible, you can use the Schur complement
One can easily show that for $$X= \begin{pmatrix} A & B \\ B^T & C \\ \end{pmatrix}$$ $X$ is positive (semi-)definite if and only if both $A$ and $S$ are positive (semi-)definite where $S=C-B^TAB^{-1}$ is the Schur complement.
In your case, $$X=\begin{pmatrix} I & A \\ A^T & 0 \\ \end{pmatrix}$$
and $S=-A^TA^{-1}$.