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On a power of matrix

Let n be a natural number such that n \geq 2. Evaluate the power

    \[\mathcal{P} = \begin{pmatrix} 1 &1 \\ 1&0 \end{pmatrix}^n\]


This is a very standard exercise in diagonalisation of matrices and there would be no reason to post it here , if it did not include the Fibonacci result. We are proving that


where F_n denotes the n – th Fibonacci number. The proof now follows with an induction on n.

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