3. A matrix S is non-negativedenite if CTSC Z O for any vector c. Prove that all covari- ance matrices are non-negativedenite. If CTSC gt; 0 the

Please help me with the following questions in linear algebra. Many thanks!

3. A matrix S is non-negative—definite if CTSC Z O for any vector c. Prove that all covari-ance matrices are non-negative—definite. If CTSC > 0 the matrix is positive definite.Show that a positive definite matrix is invertible. Also, give an example of a covariance matrix that is not positive definite. 4. Let 2 be a random vector of length n that has a M VN (0n,In) distribution. Let y : AZ + b. Derive y’s moment generating function. What is y’s distribution? 5. Let 2 be a random vector of length n that has a M VN (OMIR) distribution. LetU : ZTZ. Show that Us moment generating function is (1 — 273)_””/2 (for 1 — 215 > 0).This is the MGF for a distribution that has two names. What are they, and what are their parameters?