Question: are all positive definite matrices invertible?
Answer: If A is positive definite then A is invertible and A-1 is positive definite. ... If A is positive definite then v/Av > 0 for all v = 0, hence Av = 0 for all v = 0, hence A has full rank, hence A is invertible. For any invertible matrix A, (A-1)/ = (A/)-1.