# Math

A study conducted by the Urban Energy Commission in a large metropolitan area indicates the probabilities that homeowners within the area will use certain heating fuels or solar energy during the next 10 years as the major source of heat for their homes. The following transition matrix represents the transition probabilities from one state to another. Elec. Gas Oil Solar ElectricityNatural GasFuel OilSolar Energy 0.60 0.05 0.10 0 0.15 0.85 0.10 0.09 0.10 0.02 0.75 0.09 0.15 0.08 0.05 0.82 Among homeowners within the area, 20% currently use electricity, 35% use natural gas, 40% use oil, and 5% use solar energy as the major source of heat for their homes. In the long run, what percentage of homeowners within the area will be using solar energy as their major source of heating fuel? (Round your answer to one decimal place. Assume the trend continues.)

In time about 31.3% of people will use solar energy as their main source for heat.

We are given a transition matrix `T=([.6,.05,.1,0],[.15,.85,.1,.09],[.1,.02,.75,.09],[.15,.08,.05,.82])` The numbers reading down the columns indicate the percentage of customers "switching" to electric, natural gas, oil, and solar power (respectively). For instance, in row 3, column 2, we have .02; this means that 2% of the customers will switch from oil to natural gas. In row three, column three, we have .75, which means that 75% of customers using oil will remain using oil.

We are given an initial vector: `([.2],[.35],[.4],[.05])` which indicates that currently 20% of homes use electric, 35% use natural gas, 40% use oil, and 5% use solar power for heat.

We want to find the steady state vector to determine what percentage of homes will use solar power if the trends were to continue.

One method is to compute the steady state vector x: Tx=x.

Then Tx-x=0 and (T-I)x=0. (I is the 4x4 identity matrix.)

`T-I=([-.4,.05,.1,0],[.15,-.15,.1,.09],[.1,.02,-.25,.09],[.15,.08,.05,-.18])` We want T-I=0 so we form the augmented matrix (T-I|0) and put it in reduced row echelon form using basic matrix manipulations.

`([1,0,0,-621"/"2005],[0,1,0,-2610"/"2005],[0,0,1,-1179"/"2005],[0,0,0,0])` `x_4` is free: choose `x_4=2005`

Thus we form a basis vector `([621],[2610],[1179],[2005])` . The sum of the entries is 6415 so we can form a probability vector by dividing each entry by 6415.

`([621"/?6415],[522"/"1283],[1179"/"6415],[401"/"1283])~~([.097],[.407],[.184],[.313])`

This is the steady state vector—this gives the percentages of homes that will ultimately use each type of energy. We see that about 31.3% will use solar energy.

With a calculator we could approximate this by taking large enough powers of the transition matrix times the initial vector (any vector whose entries sum to one would work). `T^25v` with v the initial vector is accurate to 4 decimal places.