Question 6 [20 marks] At the start we discussed how thermal storage solutions are not the only possible technology of choice. Competitive options are also hydrogen storage and chemical batteries. Market research indicates that the probability to choose one of the three technologies or switch to another after each year is given by the probability matrix A: FromTl FromT2 FromT3 _ 0.8 0.1 0 ToT1 A=| 0.2 0.6 0.1 To T2 0 0.3 0.9 To T3 where the three technologies are T1: Thermal storage T2: Hydrogen storage T3: Chemical batteries Note each element in the matrix 4 is a probability. Given this, the adoption of each technology after, for example, two years can be found by ya = A Ay, for any initial state y,. Note y, represents a relative proportion of adoption of each of the three technologies, therefore its elements should add up to 1. a) [5 marks] Find the eigenvalues and eigenvectors of this matrix analytically (without the use of in-built software tools). b) [5 marks] Use your knowledge of eigenvalues, eigenvectors and the characteristic equation H_ = Ap, and the help of computational modelling, to determine if the funding distribution will reach a steady-state and if so, what that steady state funding allocation is. What are the implications for adoption of thermal storage batteries? c) [10 marks] Consider what would happen to the long-term adoption of the relative share of the three main technologies in the event that the probabilities of some of the elements in 4 are changed as outlined below. e A(1,1) is given by the probability density function 20 p20-x - J- I 0.02 e[_ ]e[_) dydx o Jo e A(23)is given by a Poisson distribution process, which represents the probability that 4 events occur, when the expected number of events occurring over the period is 6. 819