please help me solve this.

1. (12 pts) Let K and M be two m x m symmetric matrices. a Prove that matrix KM is not necessarily symmetric (To prove it, you may use prop- erties or provide a counter-example). (6 pts) b) Prove that matrix KMK is symmetric (To prove it, use properties seen in class, be precise). (6 pts) 2. (20 pts) Suppose that X is a m x n matrix and E is a m x m symmetric matrix. Consider the following matrices: A = X(XE-' X)-'X'E-1 B = X(XE- 'X)-'X' a Find the dimensions of matrices A and B. (6 pts) b) Prove that matrix A is idempotent. (8pts) c) Calculate (and simplify) AB and (B'A')' (show detailed steps). (6 pts) 3. (40 pts) Consider the system of linear equations:3351 +5172 2 7+21L'3 $1 21172 = 4+2l'2 $1 $2 + $3 = 10 331 a) Write the system in matrix form as A3 = d where :3 = 3:2 .(5 pts) 333 b) Calculate the determinant of A. Is A non-singular? Is it invertible? What is the rank of A? Comment. (7 pts) (0) Solve for 3: using the inverse matrix, i.e. a: = Ald. Show your work. (9 pts) (d) Solve for :1: using the Cramer's rule. Show your work. (9 pts) (e) Consider the system of yA' = d' where y is an unknown 1 X 3 vector and A and d are the same as above. Find y. (5 pts) 0 (f) Suppose that Arr = 0 where A and a: are given as above but the vector of the 0 constant terms is zero. What is the solution in this case? (5 pts) 4. (28 pts) Consider the following national-income model: Y = C+ lo+ Go C = atb(Y-T) a > 0 and 0 0 and 0