Help with a difficult question.

Problem 4. An n x n symmetric matrix is said to be positive definite if Ax . > > 0 for all nonzero x. (1) Prove that all eigenvalues of a positive definite matrix are positive. (2) Conversely, prove that if all the eigenvalues of a symmetric matrix A are positive, then A is positive definite. (Hint: use the Spectral Theorem) (3) Prove that every positive definite matrix is invertible, and its inverse is also positive definite