Please help me solve the following 8 questions so that I can compare it with my answers

Throughout this module, we have assumed that SSE/o' ~ Xn-p; however, we have never seen a proof for it! We now look at how to tackle this proof using matrix notation. We start by proving two simple theorems: Theorem 1: Let Y be an (n x 1) vector where Y ~ Nn (0, I). Next, let A be a (n x n) symmetric matrix such that we have the following two properties: . A is idempotent, and . A has rank r, i.e., rank(A) = r. We then have that Y'AY ~ x?. Note that the two properties listed for A imply that A permits the following eigen- decomposition A = WAW' where W is the orthogonal matrix created from the eigen-vectors of A (these eigen-vectors are denoted w1, ...;W, and are orthogonal to one another) and A is the (n x n) diagonal matrix formed from the eigen-values of A, i.e., A = Diag()1, 12, ..., An). However, since rank(A) = r we have dj = 1, j = 1, 2, ..., r and ); =0, j =r+ 1, ...,n, which means that we have . . . O 1 0 0 0 A = (rxr) (rx(n-r)) 0 Artl 0 0 0 0 ((n-r)xr) ((n-r)x(n-r)) . . . . . . 0 An Using the above information, answer the following questions to prove the result: 1. Suppose we define Z = W'Y. Determine E(Z) and Var(Z). (2) 2. Let Zi denote the ith value in the vector Z. State the distribution of Zi and Z?. (2) 3. Use the above two results and show that Y'AY ~ x?. (3) An extension of the previous theorem is to change the variance/covariance structure of the random vector.Theorem 2: We now define X as the (n x 1) vector where X ~ N, (0, 2), where E is defined as a positive definite matrix and so a non-singular matrix S erists such that we are allowed to write E = SS'. Next, let B be a (n x n) symmetric matrix such that we have the following two properties: . BE is idempotent, and . BE has rank r, i.e., rank(BE) = r. We then have that X'BX ~ x2. Using the above information, answer the following questions to prove the result: 4. Show that if BE is idempotent and E is positive definite (and thus non-singular), then it implies that BEB = B. (1) 5. Suppose we define Y = S-1X. Show that Y has a N, (0, I) distribution as required for Theorem 1. (2) 6. Since Y satisfies the conditions of Theorem 1, show that X'BX can be written in the form Y'AY (clearly define this new A matrix using the other quantities introduced above). Determine if this A matrix satisfies the two conditions stated in Theorem 1 for this type of matrix. (3) HINT: You may use the following two results without proof: . The trace of an idempotent matrix equals the rank of the matrix. . If A is (m x n) and B is (n x m), then tr(AB) = tr(BA). 7. Use the above results and Theorem 1 to confirm that X'BX ~ x?. (1) Now, assume we have the normal error regression model Y = XB+e, where & ~ N,. (0, 621), where the sample residuals are defined as e = (I - H) Y and where the error sum of squares is given by SSE = e'e = Y'(I - H)Y. 8. Set r = e/o so that r'r = e'e/o? = SSE/o2. Use the results of Theorem 1 and 2 to show that r'r follows a x' distribution with n - p degrees of freedom (thereby proving the result we originally wanted to show). (2) HINT: You may once again use the following result without proof: The trace of an idempotent matrix equals the rank of the matrix