The diagonal ellipses are meant to imply that the grid can go on to any number of rows and columns...

I found this matrix A:

Reasoning:

Look at what happens to a cell T[i,j](t) in the temperature matrix at least one row or column away from the edges of the matrix. After applying the formula the value in that cell becomes:

T[i,j](1) = k*(T[i,j](0) - (1/4)*(T[i-1,j](0)+T[i+1,j](0)+T[i,j-1](0)+T[i,j+1](0)))

So the temperature changes by a proportion k times the difference between the temperature of the cell and the average temperature of the cells around it. This is exactly as it should be by Newton's law of Cooling.

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