We can number the states as follows:

1:oil;   2: gas;   3: electric

With this numbering, the transition matrix is

P = 0.825 0.175 0 0.06 0.919 0.021 0.049 0 0.951 .

We must calculate the long-term matrix, P.

Using the technology (see the on-line 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 P²⁵⁶ has equal rows, so we take it as P:

P = 0.2447 0.5287 0.2266 0.2447 0.5287 0.2266 0.2447 0.5287 0.2266 .

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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