please provide step by step thank you

6. (a) Let x,, i = 1, ..., n be the observed p-dimensional random variables, and F(0) = -Et(x, -0) (x; -0). Denote E = "-'S. where S is the sample covariance matrix. Prove that F(0) = > + (@ -0)(5 -0)], where a is the sample mean vector. (b) Prove that det(F(@)) = det() det (1 + (@-0)"E-'(8 -0)). Hint: Consider the matrix A E r - 0 B = Ipxp (8 - O) 1 0 Calculate AB, and you need to find another matrix C, such that CA = 1+(-0)70 '(8-0)) You may directly use some facts in homework I if needed. (c) Prove that the sample covariance matrix is positive semidefinite. (d) Given that the sample covariance matrix is positive semidefinite, prove that all its eigenvalues are non-negative