Patterns in matrices

1. Oct 29, 2007

jballer23

K guys heres 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

Last edited: Oct 29, 2007
2. Oct 29, 2007

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.

3. Oct 29, 2007

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

4. Oct 29, 2007

jballer23

i'll see what i can do about the examples

5. Oct 29, 2007

jballer23

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

6. Oct 29, 2007

Zurtex

(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?

7. Oct 29, 2007

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)

8. Oct 29, 2007

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. thats all i got by looking at it

9. Oct 29, 2007

Zurtex

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)

10. Oct 29, 2007

jballer23

yes i do

11. Oct 29, 2007

jballer23

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

12. Oct 29, 2007

Zurtex

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

(1 1)n
(1 1)
=
Code (Text):

(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).

13. Oct 29, 2007

jballer23

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

14. Oct 30, 2007

Zurtex

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.

15. Oct 30, 2007

jballer23

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

16. Oct 30, 2007

jballer23

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

17. Oct 31, 2007