# Matrix Problem

Bashyboy

## Homework Statement

I attached the problem as a file

## The Attempt at a Solution

The way I tried to solve this was to write out a few multiplications and find a pattern. I got the right answer, but I was wondering if there was more of a precise way of doing it; or would the procedure that I used be acceptable for a first course in Discrete Mathematics?

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clamtrox
After you found the rule, try proving it using induction. Show first that it holds for n=2 and then show if it holds for n, then it also holds for n+1.

raopeng
Or use orthogonalisation(writing A in the form of $C^{-1}BC$) in which B is an orthogonal matrix. Then the multiplication can be simplified into: $A^n = C^{-1}B^{n}C$.

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Bashyboy
Thank you, Clamtrox.

Raopenq, we haven't learned about orthogonalisation.

clamtrox
Good, because the matrix is not diagonalizable :) If you want to understand a bit more what's happening here, it might be useful to write the matrix as
$$A = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) + \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$$
and then look how these two matrices multiply together.

Homework Helper
Start by multiplying out A2 and A3, what do you notice about the entry a12? Also, use the fact that matrix addition is entry wise.

Both of these should allow you to complete a full induction proof.

raopeng
oops sorry...