A inverse matrix $B^{-1}$ is it automatically positive definite?
Invertible matrices have full rank, and so, nonzero eigenvalues, which in turn implies nonzero determinant (as the product of eigenvalues).
*Considering the comments below, the answer is no. But if $B$ is positive semi-definite, the existence of its inverse imply $B$ positive definite?
Best Answer
Answer to original question:
No; just because a matrix has an inverse does not mean it is positive definite.
For example, $\pmatrix{-1&0\\0&-1}$ is its own inverse but is not positive definite.
Answer to edited (additional) question:
Yes; if a matrix is positive semi-definite and invertible, then it is positive definite.
This can be related to your correct assertion
that a matrix that is invertible has non-zero determinant so cannot have an eigenvalue of $0$.