We can represent the number of customers each month by a vector. In the given month the vector is
\[\mathbf{v}= \begin{pmatrix}300\\200\end{pmatrix}\]
.We can represent the change from month to moth by a transition matrix. The matrix must take into account that 10% of each customers change loyalty each month, so that if
\[A, \: B\]
are the number of customers of each company in the given month, the number of customers of A in the next month will be \[0.9A+0.1B\]
and the number of customers of company B will be \[0.1A+0.9B\]
.The transition matrix is
\[T=\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right)\]
. In the following month the customer vector will be \[\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right) \begin{pmatrix}300\\200\end{pmatrix}= \begin{pmatrix}290\\210\end{pmatrix}\]
.The number of customers of each company may become stable. In the long term
\[T \mathbf{v}=\mathbf{v} \rightarrow (M-I) \mathbf{v}=\mathbf{0}\]
.Hence
\[(M-I) \mathbf{v}=(\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right)-\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) ) \begin{pmatrix}A\\B\end{pmatrix} =\left( \begin{array}{cc} -0.1 & 0.1 \\ 0.1 & 0.1 \end{array} \right) \begin{pmatrix}A\\B\end{pmatrix} = \begin{pmatrix}-0.1A+0.1B\\0.1A-0.1B\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}\]
.Then
\[-0.1A+0.1B=0.1A-0.1B=0 \rightarrow A=B\]
but since \[A+B=500\]
we must take \[A=B=250\]
.In the long term the market will split evenly. Models like this are called Markov chains.