A-1-2-0-1-find-A-n-

Question Number 69954 by 20190927 last updated on 29/Sep/19

A = (\begin{matrix} 1 2 \\ 0 1 \end{matrix}) find A^{n}

Commented by mathmax by abdo last updated on 29/Sep/19

A = (\begin{matrix} 1 0 \\ 0 1 \end{matrix}) + (\begin{matrix} 0 2 \\ 0 0 \end{matrix}) = I + J

w e h a v e J^{2} = (\begin{matrix} 0 2 \\ 0 0 \end{matrix}) . (\begin{matrix} 0 2 \\ 0 0 \end{matrix}) = (\begin{matrix} 0 0 \\ 0 0 \end{matrix})

A^{n} = {(J + I)}^{n} = \sum_{k = 0}^{n} C_{n}^{k} J^{k} I^{n - k} = \sum_{k = 0}^{n} C_{n}^{k} J^{k}

= I + n J = (\begin{matrix} 1 0 \\ 0 1 \end{matrix}) + (\begin{matrix} 0 2 n \\ 0 0 \end{matrix}) = (\begin{matrix} 1 2 n \\ 0 1 \end{matrix})

Commented by kaivan.ahmadi last updated on 29/Sep/19

A^{2} = A . A = (\begin{matrix} 1 2 \\ 0 1 \end{matrix}) (\begin{matrix} 1 2 \\ 0 1 \end{matrix}) = (\begin{matrix} 1 4 \\ 0 1 \end{matrix})

A^{3} = A . A^{2} = (\begin{matrix} 1 2 \\ 0 1 \end{matrix}) (\begin{matrix} 1 4 \\ 0 1 \end{matrix}) = (\begin{matrix} 1 6 \\ 0 1 \end{matrix})

A^{4} = A . A^{3} = (\begin{matrix} 1 2 \\ 0 1 \end{matrix}) (\begin{matrix} 1 6 \\ 0 1 \end{matrix}) = (\begin{matrix} 1 8 \\ 0 1 \end{matrix})

⋮

\Rightarrow A^{n} = (\begin{matrix} 1 2 n \\ 0 1 \end{matrix})

Commented by 20190927 last updated on 29/Sep/19

thank you sir