The weather on an uninhabited island can be predicted as follows: If today is sunny,...

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The weather on an uninhabited island can be predicted as follows: • If today is sunny, then tomorrow is - sunny with a 40% chance; - cloudy with a 40% chance; - rainy with a 20% chance. If today is cloudy, then tomorrow is - sunny with a 20% chance; - cloudy with a 40% chance; - rainy with a 40% chance. . If today is rainy, then tomorrow is - sunny with a 40% chance; - cloudy with a 20% chance; - rainy with a 40% chance. Associate sunny, cloudy, and rainy with statuses 1, 2, and 3 respectively. The prediction chances can be expressed as a 3 x 3 matrix A = (a), where aj is the chance that tomorrow is state j if today is state i. (a) Let A² = (a). Show that a, is the chance that 2 days later is state j if today is state i. (b) Show that A³ = 1.24²-0.244+0.041, where I is a 3x3 identity matrix. (c) What is the chance that it is rainy 4 days later if today is sunny? The weather on an uninhabited island can be predicted as follows: • If today is sunny, then tomorrow is - sunny with a 40% chance; - cloudy with a 40% chance; - rainy with a 20% chance. If today is cloudy, then tomorrow is - sunny with a 20% chance; - cloudy with a 40% chance; - rainy with a 40% chance. . If today is rainy, then tomorrow is - sunny with a 40% chance; - cloudy with a 20% chance; - rainy with a 40% chance. Associate sunny, cloudy, and rainy with statuses 1, 2, and 3 respectively. The prediction chances can be expressed as a 3 x 3 matrix A = (a), where aj is the chance that tomorrow is state j if today is state i. (a) Let A² = (a). Show that a, is the chance that 2 days later is state j if today is state i. (b) Show that A³ = 1.24²-0.244+0.041, where I is a 3x3 identity matrix. (c) What is the chance that it is rainy 4 days later if today is sunny?