I found this matrix C:

The interesting thing about C is when you do this operation:

(matrix multiplication)

For the example matrix I gave A, this gives:

The sum of all of these entries is exactly the same as the sum of all the entries in matrix A!

If you look at matrix A and the resultant matrix you can see that every cell gets closer to the average value of all the entries in the matrix, which is what you would expect from a grid of temperatures settling into an average temperature.

Let's look at the bottom right cell in A, that starts off at 7 and goes to 4.875, another application of the matrix multiplication with C gives:

7

4.875

4.1406

3.83

3.673

3.584

3.531

3.498

3.478

...

You can see it is is sort of exponentially decaying to the average temperature of 3.444

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