TITLE: Inverse of a Positive Definite
QUESTION [37 upvotes]: Let K be nonsingular symmetric matrix, prove that if K is a positive
  definite so is $K^{-1}$ .

My attempt:
I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next.

REPLY [13 votes]: K is positive definite so all its eigenvalue are positive. The eigenvalues of $K^{-1}$ are inverse of eigenvalues of K, i.e., $\lambda_i (K^{-1}) = \frac{1}{\lambda_i (K)}$ which implies that it is a positive definite matrix.