Finding a Solution to a Discrete Math Problem: Is Precise Necessary?

[Bashyboy]

Homework Statement

I attached the problem as a file

Homework Equations

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?

 

[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.

 

[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.

 

[STEMucator]

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...