Patterns in Matrices: P^n and S^n Calculations

[jballer23]

K guys here's the problem

P= (3 1
1 3)

S=(4 2
2 4)

Calculate P^n and S^n for other values of n and describe any patterns you see.
I tried this one for about an hour and got a little bit. I just want to see what you can get out of it. Maybe I missed something. Please Help! thanks

 

[Zurtex]

What values of n did you calculate it for? Can you show us a few examples and post anything if you anything, or not if you don't.

 

[jballer23]

i calculated it out for 1,2,3,4,and 5 its really hard to post on my computer. do you have any ideas for finding a general form? because that is the basis of the problem

 

[jballer23]

i'll see what i can do about the examples

 

[jballer23]

P^3= (36 28
28 36)
P^4= (136 120
120 136)
P^5= (528 496
469 528)

 

[Zurtex]

i calculated it out for 1,2,3,4,and 5 its really hard to post on my computer. do you have any ideas for finding a general form? because that is the basis of the problem

(3 1)²
(1 3)
=
(10 6)
(6 10)


(3 1)³
(1 3)
=
(36 28)
(28 36)


(3 1)⁴
(1 3)
=
(136 120)
(120 136)


Do you not spot a pattern?

Are you familliar with proof by induction?

 

[jballer23]

S^2= (20 16
20 16
S^3= (112 104
112 104)
S^4= (656 640
640 656)
S^5= (3904 3872
3872 3904)

 

[jballer23]

no I'm not sorry I'm trying to learn this. its an assignment my teacher gave us and told us to run with. i saw one pattern but i don't really know how to explain it. i noticed that the first term in each matrix differed from the second term by 2^n. that's all i got by looking at it

 

[Zurtex]

no I'm not sorry I'm trying to learn this. its an assignment my teacher gave us and told us to run with. i saw one pattern but i don't really know how to explain it. i noticed that the first term in each matrix differed from the second term by 2^n. that's all i got by looking at it

That's quite cool, do you know how to summate terms like this:

$$\sum_{x=1}^n x$$
?

(Not this particular example, but that sort of style of summation)

 

[jballer23]

yes i do

 

[jballer23]

yes she has taught us that but i don't know what that has to do with it?

 

[Zurtex]

yes i do

Think about trying to multiply the matrix "n times then". Perhaps start with an easy example then like:

(1 1)ⁿ
(1 1)
=

(1 1) (1 1) (1 1) ... (1 1)
(1 1) (1 1) (1 1)     (1 1)

(Try actually writting what's happening in each element, you should get a bit of a long sum, that you can calculate).

 

[jballer23]

ok i did that but I'm still not getting how to work that with my original problem

 

[Zurtex]

ok i did that but I'm still not getting how to work that with my original problem

Well it's the same princaple, if you get a summation form in each of the element, you've worked out what it is, more over you may be able to put it in a closed form if you understand how to do the summations.

 

[jballer23]

ok thank you, i'll try that today I'm pretty sure i'll be able to work it out now. That helped a lot.

 

[jballer23]

hey i couldn't find any patterns that way. did you find anything?

 

[Zurtex]

hey i couldn't find any patterns that way. did you find anything?

Yeah, I worked them both out pretty quickly, just trying to help you along rather than give the answer. I don't know how else to help you without just saying the answer :/

 

[jballer23]

ok well i turned in the paper today hopefully it is right. the general form i came up with was like a scalar or 2^(n-1) (k^n+1 k^n-1)
(k^n-1 k^n+1)