How to solve these two questions of algebra?

1.

(2 marks) Let X be the random variable with the probability distribution given by P(X=k)=i gk' 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 = CO CT 6 and let the eigenvalues of A be 1 and 12 . Then *1 + 12 = and X1 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 correspodeing to the eigenvalue 5 is where q = G D. c) Suppose that C is a 2 X 2 matrix, with eigenvalues 3 and 1 corresponding to the eigenvectors (1 ) and (2 ) respectively. Then C4 a o C d where a = A D, 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 co co 16 eigenvalue 3. 1 You are also given that 2 4 + 2 3 18 4 Then A3 2 S 18 where r = S and t=