Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?
First, if a matrix is positive semidefinite then it can have eigenvalues equal to zero, in which case it is singular.
If it is positive definite (using the most common definition, i.e. symmetric and with positive eigenvalues) then the answer is yes since the eigenvalues of $A^{-1}$ are the reciprocals of the eigenvalues of $A$.