Answer Happy

Three heating oil suppliers identified as C1, C2 and C3 share the market in a given region. Each year, part of each company's clientele leaves to do business with one or other of the other two companies. The following matrix is the transition matrix for this market: (0,4 0,5 0,1) 0,1 M = 0,5 0,4 0,5 0,1 0,1 0,4 where mij is the proportion of company Ci's customer base that moves to the Cj company every year. If the variables x and y respectively represent the market shares (in percentage) of companies C1 and C2 at the end of a year, the following equation represents the equilibrium situation: (0,4 0,5 0,1 0,5 0,4 0,5 0,1 0,1 0,4 y y (₁ -:--)-(, -_-) 1-x-y
a) Write the system of linear equations with two unknowns equivalent to this matrix equation. b) Solve this system. c) What will be the market shares (in percentage) of each of the three companies when equilibrium is reached?