If we number the states as 1: Oil;  2: Gas;   3: Electric, we get the transition matrix from the given table:

P = 0.825 0.175 0 0.060 0.919 0.021 0.049 0 0.951 .

This is the one-step transition matrix. To obtain the long-term behavior using technology, we need to take higher and higher powers of P until it stops changing. On a TI83 graphing calculator, you can compute P² and keep squaring the answer as follows: First enter the matrix P under the name [A]. Then do the following:

[A]   [ENTER]	Displays the matrix P
^2   [ENTER]	Squares the current matrix (P)
[ENTER]	Repeats the above instruction: squares the current matrix (P2), giving P4
[ENTER]	Gives P8
...	...

Alternatively, if you use the on-line matrix algebra tool, you can enter P in the input window, and then type "P^2" or "P^4" or "P^8" ... in the formula window.

Ultimately, the powers of P will stop changing (to 6 decimal places) when we reach P²⁵⁶:

P256 = 0.244713 0.528701 0.226586 0.244713 0.528701 0.226586 0.244713 0.528701 0.226586 .

Thus, if current trends continue, 24.47% of all US homes will use oil heat, 52.87% will use gas heat, and 22.66% will use electic heat in the long term.

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