matrices powers: needing to find pattern in X, X^2, X^3,....

[gyrag]

hi! Im having trouble with this question about matrices

the matrix X =

(4 2)
(2 4)

X to the power of two =

2 (10 8 ) or
...(8 10)

4(5 4)
..(4 5)

X to the power of three =

2 (56 52)
...(52 56)

or 8 (14 13)
.......(13 14)

and so on...

I have to figure out an "pattern" so i can have a general formula for the scalar (which ive figured out) and the elements within the matrix.. which i cant seem to find... it has something to do with 3 to the power of n. Ive tried lots of combinations but none of them seem to fit.

 

[soroban]

Re: matrices powers

Hello, gyrag!

This is not an easy one . . .

\(\displaystyle X\:=\:\begin{pmatrix} 4 & 2 \\ \\ \\ 2 & 4\end{pmatrix} \:=\:2\cdot\begin{pmatrix}2 & 1 \\ \\ \\ 1 & 2\end{pmatrix}\)

\(\displaystyle X^2\:=\:2^2\cdot\begin{pmatrix}5 & 4\\ \\ \\4&5\end{pmatrix}\)

\(\displaystyle X^3\:=\:2^3\cdot\begin{pmatrix}14 & 13 \\ \\ \\ 13 & 14\end{pmatrix}\)

\(\displaystyle X^4\:=\:2^4\cdot\begin{pmatrix}41 & 40 \\ \\ \\ 40 & 41\end{pmatrix}\)

\(\displaystyle X^5\:=\:2^5\cdot\begin{pmatrix}122 & 121 \\ \\ \\ 121 & 122\end{pmatrix}\)


The matrices are of the form: \(\displaystyle \:2^n\,\cdot\,\begin{pmatrix}a & a-1 \\ \\ \\ a-1 & a\end{pmatrix}\)

. . where: \(\displaystyle a \:=\:2,\,5,\,14,\,41,\,122,\,\cdots\)


I found the formula for the \(\displaystyle a's:\)
. . \(\displaystyle \L a_n\:=\:\frac{3^n\,+\,1}{2}\)
Therefore: \(\displaystyle \L\:X^n \:=\:2^n\cdot\begin{pmatrix}\frac{3^n\,+\,1}{2} & \;\frac{3^n\,-\,1}{2} \\ \\ \\ \frac{3^n\,-\,1}{2} & \frac{3^n\,+\,1}{2} \end{pmatrix}\)

 

[gyrag]

thanks so much!

also if i may ask, how would you find the "domain" of the matrix.. is there any specific method or just trials on the calculator

Thanks again