Fibonacci Matrix HELP

[famine]

Stuck with an assignment question. Explanation or any help would be greatly appreciated .

F= 1 0
1 1
Sorry if its looks average but thats my attempt at drawing the matrix.

It asked to compute the F^n where n=2,3,10 and 40

Also asks to find a determinant how do you go about doing this? (determinant of 40)

Thanks.

 

[soroban]

Hello, famine!

Have you learned Matrix Multiplication and Determinants?

\(\displaystyle \text{Given: }\:F \:=\:\begin{pmatrix}1&0\\1&1\end{pmatrix}\)

\(\displaystyle \text{Compute }F^n\text{ where }n=2,3,10, 40.\)

\(\displaystyle F^2 \;=\;\begin{pmatrix}1&0\\1&1\end{pmatrix}\cdot \begin{pmatrix}1&0\\1&1\end{pmatrix}\;=\; \begin{pmatrix}1&0\\2&1 \end{pmatrix} \)

\(\displaystyle F^3 \;=\;\begin{pmatrix}1&0\\2&1\end{pmatrix}\cdot \begin{pmatrix}1&0\\1&1\end{pmatrix}\;=\; \begin{pmatrix}1&0\\3&1 \end{pmatrix} \)

\(\displaystyle \text{In general: }\:F^n \:=\:\begin{pmatrix}1&0\\n&0\end{pmatrix}\)

\(\displaystyle F^{10} \;=\;\begin{pmatrix}1&0\\10&1\end{pmatrix}\)

\(\displaystyle F^{40} \;=\;\begin{pmatrix}1&0\\40&1\end{pmatrix}\)

\(\displaystyle \text{Find the determinant of }F^{40}.\)

\(\displaystyle \begin{vmatrix}1&0\\40&1\end{vmatrix} \;=\;(1)(1) - (0)(40) \;=\;1\)


This problem does not involve the Fibonacci Sequence.

 

[HallsofIvy]

I suppose you could sit and stare at the problem until the answer magically appeared on your paper. (That has never worked for me but others seem to like it.)

Or you could ask some one else to do it for you. (Oh, you've already thought of that, haven't you.)

As an absolute last resort, you might try actually multiplying the matrix by itself! You probably don't want to do that 40 times but doing it a few times might show you a possible pattern. I see that Soroban has already done that for you. If I were your teacher, I would insist that you show that this pattern is correct. You can do that using "proof by induction".

 

[mmm4444bot]

F= 1 0
1 1
Sorry if its (sic) looks average but thats (sic) my attempt at drawing the matrix.

You do not need to "draw" that matrix, but if you insist please use the code tags.

Simple matrices may be texted like this:

F =
[1, 0]
[1, 1]

Cheers :cool: