1. Woolies, Coles and ALDI are fighting for market share in Australia. Every year: Woolies loses 30% of its customers to Coles and 40% of its customers to ALDI. Coles loses 40% of its customers to Woolies and 20% of its customers to ALDI. ALDI loses 50% of its customers to Woolies and 30% to Coles. Let n, Yn, Zn denote the market shares of Woolies, Coles and ALDI after n years. Currently the respective market shares are 40%, 40% and 20% i.e. = 0.4 20 0.2 Note the column vector here adds up to 1 and this is because in this model, we're assuming there is no other competitor besides Woolies, Coles and ALDI. 21 0.3 0.4 0.5\ We have Y1 Ayo where A = 0.3 0.4 0.3 21 0.4 0.2 0.2) In TO and Yn = An Yo Zn 20 B has three eigenvectors are 1, -0.2, 0.1 respectively. Explain why as n o, a/. In V2= V3= and corresponding eigenvalues 2 Yn kV1. Zn Where k is a scalar b/ Assuming that in future no company will come forth to compete with Woolies, Coles, and ALDI, find the value of k. Hence give the market shares of Woolies, Coles, and ALDI in the long term. c/. If we wanted to compute the exact market shares of Woolies, Coles and ALDI 20 years from now according to this model, what would be the most efficient method? You're not expected to actually do this computation; simply give a brief outline of what steps you would take