I see there is a bit of confusion, so I'll try to explain better.
A matrix $A$ is positive semidefinite (p.sd.) if it is symmetric and all its eigenvalues are $\ge 0$.
A matrix $A$ is positive definite (p.d.) if it is symmetric and all its eigenvalues are $>0$.
This means that every p.d. matrix is also a p.sd. matrix. The set of positive semidefinite matrices contains the set of positive definite matrices, but there are some p.sd. matrices that are not p.d., like the matrix that has all elements equal to zero.
As an analogy, you can think about the real and integer numbers: all integer numbers are real numbers, because the set of real numbers contains the set of integer numbers, but there are real numbers, such as $\sqrt 2$ that are not integer.
The most common example is the identity matrix $I$: all its eigenvalues are $1>0$, so it is a p.d. matrix, but $1\ge 0$ so it is also a p.sd. matrix. This means that $I$ is a p.sd. matrix and it is invertible.
Best Answer
The $n \times n$ zero matrix is positive semidefinite and negative semidefinite.