Question

Calculați A^n, n aparține N stelat unde A=
(1 0 1
0 1 0
1 0 1)​

Answer 1

$A^2=\left[\begin{array}{ccc}2&0&2\\0&1&0\\2&0&2\end{array}\right] \\A^3=A^2A=\left[\begin{array}{ccc}4&0&4\\0&1&0\\4&0&4\end{array}\right]$

Prin inducție obținem că:

$A^n=\left[\begin{array}{ccc}2^{n-1}&0&2^{n-1}\\0&1&0\\2^{n-1}&0&2^{n-1}\end{array}\right]$  ∀ n ∈$N^{*}$