How to solve these two questions of algebra?

1.

(2 marks) Let X be the random variable with the probability distribution given by c = 976' fork=0,1,2,.... (a) The value of c in exact form is (b) Furthermore if d is an integer such then the value of d is (6 marks) a) Let A = 6 and let the eigenvalues of A be 1 and 12 . Then *1 + 12 = and 1 12 = b) Let B = -10 12 -15 17 Given that B has eigenvalues 2 and 5, an eigenvector corresponding to the eigenvalue 2 is , where P = and an eigenvector correspodeng to the eigenvalue 5 is where q = c) Suppose that C is a 2 x 2 matrix, with eigenvalues 3 and 1 corresponding to the eigenvectors ( 1 ) and respectively. Then C4 = a where a = 4 0,b= C : and d =4 d) Suppose that the matrix A has the eigenvector -4 corresponding to the eigenvalue -1, and the eigenvector corresponding to the 16 eigenvalue 3. 6 1 You are also given that 2 = 4 -1 + 2 3 18 4 6 Then A3 2 S 18 where r = S = and t =