A classic problem that can be handled with matrices involves temperature distribution—how the different temperatures at points on a surface are affected by the temperatures closest to them. This is especially important when you can’t take all the temperature measurements throughout the surface.
Say that you have a large, flat, aluminum roof and want to know the approximate temperatures at different positions on the roof. You can’t go out on the roof (ouch, hot, wobbly), but you can measure along the edges. You also know of a property that says the temperature at a point on the surface is the average of the four closest points on a grid drawn over the surface. The following figure represents the points on your roof.
How hot are the spots on the roof?The temperature at spot x is the average of the temperatures of the four closest points. To the left, the temperature is 120 degrees; above the point, the temperature is 80 degrees; to the right, it’s y degrees; and below, it’s z degrees. So the temperature at point x is the average of the four measures.
Creating the temperatures at the other three points, you have
Multiplying each equation by 4 and arranging the terms in the same order, you have
Now, solve the system for x, y, z, and w. Using an augmented matrix, you get
You go through the steps to put it in reduced-echelon form and get
So the temperature at x = 98.75, at y = 91.25, at z = 103.75, and at w = 96.25 degrees, as shown here.
The average temperatures on the roof.