Home » Uncategorized » On a power of matrix

On a power of matrix

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

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

Solution

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

    \[\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}\]

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

Read more

Leave a comment

Donate to Tolaso Network