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 0.06 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.