I have done the first five questions I need help from f to i only

1. Woolies, Coles and ALDI are fighting for market share in Australia. Every year: . Woolies loses 30% of it 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 In, Un, Zn denote the market shares of Woolies, Coles and ALDI after n years. Currently the respec ive market shares are 40%, 40% and 20% i.e. yo zo 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. X1 TO 0.3 0.4 0.5 (a) Explain why y1 = A yo where A = 0.3 0.4 0.3 21 20 0.4 0.2 0.2 TO (b) Explain why yn = An yo Zn (c) Show that A has eigenvectors V1 V2 = V3 = -3 and find the corresponding eigenvalues 1, 12, 13. (d) Explain why v1, V2, V3 form a basis for 3. (e Hence explain why we can find scalars k1, k2, k3 E R such that TO yo = kiv1 + k2v2 + k3V3, zo without actually finding k1, k2, k3. (f ) Use parts (b) and (e) to therefore show that: In Un = kidiv1+ k2\\zv2 + k3\\g V3- Zn (g) Explain why as n -+ 0o, In yn -+ kivi. Zn (h) Assuming that in future no company will come forth to compete with Woolies, Coles and ALDI, find the value of 1. Hence give the market shares of Woolies, Coles and ALDI in the long term. (i) 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