Let's assume companies A$,$ B$,$ and C make the same product and at the beginning of the

year have $20\%,35\%,$ and $45\%$ of the market share respectively. At the end of each year,

a$.$ company A keeps $70\%$ of its own market share, and companies B and C gain $10\%$ and

$20\%$ of its market share

b$.$ company B keeps $85\%$ of its own market share, and companies A and C gain $5\%$ and

$10\%$ of its market share

c$.$ company C keeps $80\%$ of its own market share, and companies A and B gain $15\%$ and

$5\%$ of its market share

We can define market share transformation matrix and initial market share vector as

$A=([0\mathrm{.}7,0\mathrm{.}05,0\mathrm{.}15],[0\mathrm{.}1,0\mathrm{.}85,0\mathrm{.}05],[0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}8]),S=([20],[35],[45])$

The market share of these companies after one year is $AS.$

What is the market share of these companies after $3,5,$ and $10$ years?

What is the steady state market share of these companies?

Find the eigenvalues of A and explain how you can find the steady state market share of

these companies using eigenvalue and eigenvector.

Use MATLAB for solving this problem.