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