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Please do look at the results of taking that matrix to different powers: https://www.wolframalpha.com/input?i=%5B%5B1%2C+1%5D%2C+%5B1...

What is this matrix doing? When you left multiply it, you're summing the elements of the left column and putting them in the top left (next fib number). Then you're moving the current number to bottom left (and top right). Incidentally, you move the previous to bottom right.

That's interesting!! It's very simple

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Yes, indeed. If you see the thirty three miniatures book, it gives the matrix exponentiation formula for computing Fibonacci numbers, without any explanation behind it. To write the article, I tried to figure out ways it can be explained. And this pattern you suggested shows up nicely even when you express successive Fibonacci numbers in terms of the base cases F0 and F1. Which ultimately lead me to realize, the coefficients are the elements of the matrix.

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