We can number the states as follows:
With this numbering, the transition matrix is

We must calculate the longterm matrix, P^{}.
Using the technology (see the online matrix algebra tool for an example), we approximate P^{} by calculating higher and higher powers of P. (Using powers of 2 as the exponent speeds up the calculation.) If you experiment, rounding the answers to 4 decimal places, you will find that the matrix P^{256} has equal rows, so we take it as P^{}:

Thus, 24.47% of of all homes will be using oil, 52.87% will be using gas, and 22.66% will be using electricity for heating.
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