Question 2 [It'- marks] In a certain shopping mall, one of three messages is broadcast every hour. The rst message, called message a, welcomes the patrons to the mall. The second message, called message I}, reminds the patrons to observe social distancing. The third message, called message c, announces the promotions of the day. The probability that each of the message is selected to be broadcast for the hour depends only on what was broadcast in the previous hour. {i} If message a was broadcast in the previous hour, then for the current hour, message a will be broadcast again with probability 0.3, message tr will be broadcast with probability {1.3, and message c will be broadcast with probability {1.4. [ii] If message it was broadcast in the previous hour, then for the current hour, message a will be broadcast with probability [L3, message I} will be broadcast again with probability [1.3, and message n: will be broadcast with probability 0.4. {iii} If message c was broadcast in the previous hour, then for the current hour, message a will be broadcast with probability 13.4, message I} will be broadcast with probability [1.4, and message s will be broadcast again with probability {1.2. Suppose the welcome message [message a} is being broadcast currently. Let an, on, cn be the probability that message a, b, c would be broadcast after in. hours, respectively. {a} [2 marks] Find a matrix A such that an+l an. bn+l = A bn , 511+]. En forallna. {b} [3 marks] Let A be the matrix dened in {a}. Find an orthogonal matrix P and a diagonal matrix D such that A = PDPT. The entries of the matrix P must be exact, leave your answer in surds if necessary. [You may use MATLAE to aid in your computations, but you must show your workings clearly. Marks will be awarded to workings] (c) [4 marks] Compute en, n, cn each as a function etc, for u 3 1. Note that the welcome message is being broadcasted currently. {d} [2 marks] 1What are the probabilities on, It\