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November 1st, 2017, 05:12 PM   #1
ZMD
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Power Formula

Power Formula for A=$\begin{pmatrix} \Lambda & 1 & 0\\ 0 & \Lambda & 1\\ 0 & 0 & 1 \end{pmatrix}^n$

Last edited by ZMD; November 1st, 2017 at 05:16 PM.
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November 1st, 2017, 05:41 PM   #2
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Originally Posted by ZMD View Post
Power Formula for A=$\begin{pmatrix} \Lambda & 1 & 0\\ 0 & \Lambda & 1\\ 0 & 0 & 1 \end{pmatrix}^n$
I don't know what you mean by the term "power formula." But it's fairly easy to spot the pattern for $\displaystyle A^n$ by working out a few terms if that's what you need. I get
$\displaystyle A^n = \left ( \begin{matrix} \Lambda ^n & n \Lambda ^{n - 1} & 1 + 2 \Lambda + \dots + (n - 1) \Lambda ^{n - 2} \\ 0 & \Lambda ^n & 1 + \Lambda + \Lambda ^2 + \dots + \Lambda ^{n - 1} \\ 0 & 0 & 1 \end{matrix} \right )$

You can use induction to prove this formula.

Did that answer your question?

-Dan
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