Please explain how to solve step by step and name any rules or theorems used.

The following is from Chiang's fundamental mathematical economics.

1. Given A = [_; ; ]. 8 - [8 -8]. andc =[! " ;]. find A:, 8', and C'. 2. Use the matrices given in Prob. 1 to verify that (0) (A + B)' = A' + B' (b) (AC)' = C'A' 3. Generalize the result (4.11) to the case of a product of three matrices by proving that, for any conformable matrices A, B, and C, the equation (ABC)' = C'B'A' holds. 4. Given the following four matrices, test whether any one of them is the inverse of another: D=[8 3] = = [6 8] #= [6 7] -[$ 1 ] 5. Generalize the result (4.14) by proving that, for any conformable nonsingular matrices A, B, and C, the equation (ABC) 1 = C-18-1A-1 holds. 6. Let A = 1 - X(X'X)-1X'. (0) Must A be square? Must (X'X) be square? Must X be square? (b) Show that matrix A is idempotent. [Note: If X' and X are not square, it is inappro- priate to apply (4.14).] 4. Prove that for any two scalars g and k (a) k(A + B) = KA +KB (b) (g + k)A = gA + KA (Note: To prove a result, you cannot use specific examples.) 5. For (a) through (d) find C = AB. (a ) A = 12 14 20 B = 0 2 ( b) A = B = 2 8 5 6 7 7 11 ( C) A = 2 9 8 = 12 3 4 6 10 6 10 (0) A = 2 5 9 B = 11 3 2 9 (e) Find (@) C = AB, and (ii) D = BA, if AN A = B = [3 6 -2] 6. Prove that (A + B)(C + D) = AC - AD + BC + BD. 7. If the matrix A in Example 5 had all its four elements nonzero, would x'Ax still give a weighted sum of squares? Would the associative law still apply? 8. Name some situations or contexts where the notion of a weighted or unweighted sum of squares may be relevant