Question 2 a) Calculate the eigenvalues and eigenvectors of the matrix A: [25) (X + 5) (Y + 20) (Y + 15) A = [x+ 10) -7 where X and Y are the 2nd last and last digits of your URN respectively. For example, for the URN 6835725, x = 2 and Y = 5. Please write your values for X and Y at the top of your solution. b) Consider a 3x3 matrix B: [1+X B = 11 +Y Y- 4 15 where X and Y are based on your URN, as in part (a) The eigenvalues of B are: 2, = 15, d2 = 1 + X, 23 = 11 + Y and the corresponding eigenvectors are: %3D [1/(X - 14)] X = [1/(X Y 10)] X2 = and X = -1 We write a diagonal matrix D such that the elements along the main diagonal are the eigenvalues of B: [15 0 0 D =0 1+ X 11 +Yl We also write a matrix P such that the columns of Pare the eigenvectors of B: [1/(X-14) 1 1/(X -Y - 10)] P = -1 I) Compute P (8] i) Confirm that B = PDP. (Writing a matrix in this form is called matrix diagonalization since D is a diagonal matrix). [5] Acti H) One of the uses of the identity in i) is to compute large powers of a matrix. Using the identity in i), calculate B" (do not multiply out any exponents in the elements of the resulting matrix - eg. If you get 7 in an element do not write it as 117649, leave it as 7). Gto [71