soso Posted November 28, 2008 Report Share Posted November 28, 2008 (edited) Hi, In this portfolio, we have to investigate matrix binomials. X=(1,1,1,1) Y=(1,-1,-1,1)Let A=aX and B=bY, where a and b are constants.Use different values of a and b to calculate: A^2, A^3, A^4 ; B^2, B^3, B^4By considering integer powers of A and B, find the expressions for A^n, B^n, (A+B)^n----this is where i mess up, i cannot find a general term for (A+B)^n...can anyone give me some tips or something?further more, i have tried numbers for a and b that are the same, that's easy, but when a and b are different numbers, i mess up..i cant get anything ...thanks Edited November 28, 2008 by soso Reply Link to post Share on other sites More sharing options...
ezex Posted November 28, 2008 Report Share Posted November 28, 2008 (edited) Wow I can't believe they're showing you guy this stuff. I don't think you realized what you were ALMOST doing here, Eigenvalues and Eigenvectors. It's all linear algebra topics, look them up if you want but basically it's when you have a matrix and you multiply it by another and get, as your answer, the same matrix times a constant (so A is the same as a constant times your matrix). I'm not sure exactly how you would go around solving this problem without basic eigenvalue rules, but just if you hadnt realized, what you are doing is finding constant multiples of your matrix, so what you want to do is show that A^2 is the same as saying (a)^2 times X^2, but you can't square a non square matrix, so you have to transpose it and get a column vector times a row vector in which case you would get that A^2, for a=1, is 4 and A^3 is A times A^2, but A^2 is a constant so you go back to your original form of your equation, constant times matrix, so is 4 times A and A is (1,1,1,1). And for a = 2, A^2 is either (2,2,2,2) times the column of that which would give you 16, or do (1,1,1,1) squared times 2 squared which gives you 4 times 4 = 16. For B its the same and so find the pattern, that i wont give you or else i solved the entire thing for you, and you'll see that the constants are the only things being multiplied.Oh and you're not supposed to use different values for a and b...at least it doesn't seem like you should. Edited November 28, 2008 by ezex Reply Link to post Share on other sites More sharing options...
soso Posted November 28, 2008 Report Share Posted November 28, 2008 Yes, i already got A^n and B^n, the thing is, that if you use different values for a and b for (A+B)^n, then i get messed up...whereas if you use the same values, i can get the general term Reply Link to post Share on other sites More sharing options...
Toth Posted December 12, 2008 Report Share Posted December 12, 2008 Here's a tip that made my portfolio so much easier:Note how matrices X and Y multiply together to make a matrix that is completely zeroes.Do a little research on this special matrix, and find out why it's so significant. It'll help you out.And yes, you do use different values for constants a and b, your general statement should sort of resemble the ones created for X^n, A^n, Y^n, and B^n. Reply Link to post Share on other sites More sharing options...
Petitchat Posted December 28, 2008 Report Share Posted December 28, 2008 hey i'm doing the same matrix binomial portfolio that you are doing and I'm Lost. Can you please help me ? i dont get the part where it says "BY considering integer powers of X and Y, find expressions for x^n, y^n, (X+Y)^n. Please help me.yes, i am having trouble on the same part. specifically, what would the expressions look like? Reply Link to post Share on other sites More sharing options...
c_m Posted December 29, 2008 Report Share Posted December 29, 2008 Heyy I am also doing this one! Do either of you have an example that I would be able to see, I have a teacher who is new to IB and did not explain it very well. It is due in less than a week thx Reply Link to post Share on other sites More sharing options...
vp90 Posted December 30, 2008 Report Share Posted December 30, 2008 Hey! i'm also doing the portfolio n have almost finished, i'm posting the expressions i obtained hope they help.X^n = 2^(n-1) * (1,1, 1,1)Y^n = 2^(n-1) * (1,-1, -1,1)(X+Y)^n = 2^(n-1) * (1,0, 0,1)A^n=2^(n-1) * a^n * (1,1, 1,1)B^n=2^(n-1) * b^n * (1,-1, -1,1)(A+B)^n = 2^(n-1) * (a^n * (1,1, + b^n * (1,-1, ) 1,1) -1,1)M^n = 2^(n-1) * (a^n * (1,1, + b^n * (1,-1, ) 1,1) -1,1)The important thing to show is how you got to these, and it is not necessary that you get the same general statements as above they may vary. Reply Link to post Share on other sites More sharing options...
vp90 Posted December 30, 2008 Report Share Posted December 30, 2008 Mod Edit: You are not allowed to post the expressions. That is against IB policy -Mike Reply Link to post Share on other sites More sharing options...
Chrisander Posted January 29, 2009 Report Share Posted January 29, 2009 Im desperate right now. Im handing in my portfolio tomorrow and Im not even half way. Can anyone help? Reply Link to post Share on other sites More sharing options...
moneyfaery Posted January 30, 2009 Report Share Posted January 30, 2009 What is your question? Reply Link to post Share on other sites More sharing options...
Chrisander Posted January 30, 2009 Report Share Posted January 30, 2009 What is your question?Have you seen the question about Matrix Binomials for this year? Its this part Im struggeling with:By considering integer powers of A and B, find expressions for A^n, B^n, (A+B)^nIm not able to find expression for the last one (A+B)^n Reply Link to post Share on other sites More sharing options...
M. Chinchilla Posted February 1, 2009 Report Share Posted February 1, 2009 Yes, i already got A^n and B^n, the thing is, that if you use different values for a and b for (A+B)^n, then i get messed up...whereas if you use the same values, i can get the general termhey, i figured A^n and B^n but im kinda stuck with (A+B)^n. help please. Reply Link to post Share on other sites More sharing options...
Toth Posted February 3, 2009 Report Share Posted February 3, 2009 To find out (A+B)^n, you need to do research on the significance of the identity matrix. Then you can continue your formula, because the identity matrix will nullify all terms except for the first and the last.Then you can easily make your general statement. Reply Link to post Share on other sites More sharing options...
Toth Posted February 3, 2009 Report Share Posted February 3, 2009 Find out what the identity matrix is! It's crucial to answering your portfolio!Plus it will make your life 10x easier! Reply Link to post Share on other sites More sharing options...
ibkillsme Posted February 12, 2009 Report Share Posted February 12, 2009 hey toth, do you mind elaborating on the significance of the identity matrix? im also pretty far, but still stuck with finding an expression for (A+B)^n. how does the identity matrix help?Secondly, afterwards they ask to show that M=A+B and M²=A²+B². I showed this with different integer values of a and b, but what is the connection we are supposed to make?i would really appreciate your help! Reply Link to post Share on other sites More sharing options...
TI84 Silver Edition Posted February 25, 2009 Report Share Posted February 25, 2009 I am doing this right now for Math IB, it is soo confusing because the instructions are so vague. Can someone make suggestions about how I can go beyond what was written in the instructions? Reply Link to post Share on other sites More sharing options...
glucose Posted March 2, 2009 Report Share Posted March 2, 2009 i'm still confused on how to get (A+B)^n, can anyone help? how do u find the identity matrix and what does it have to do with solving this question? Reply Link to post Share on other sites More sharing options...
Hamtaroo Posted March 4, 2009 Report Share Posted March 4, 2009 i'm still confused on how to get (A+B)^n, can anyone help? how do u find the identity matrix and what does it have to do with solving this question?Try expanding the expression, then finding what AB is...Then it should be easy if you know A^n and B^n. Reply Link to post Share on other sites More sharing options...
courtney Posted March 7, 2009 Report Share Posted March 7, 2009 (edited) i was wondering if anyone has any idea what we're supposed to do with this portfolio for the informal proof. from what i've been told, not too many students get credit for that part of the portfolio, but since i have a couple more days, i figure i should at least try. any suggestions?also, what happens when n=0 in the general statement? does it make it untrue, or would 0 be a limit? Edited March 8, 2009 by courtney Reply Link to post Share on other sites More sharing options...
ibastudent Posted March 15, 2009 Report Share Posted March 15, 2009 Hi,I still don't understand how the identity matrix will help since there are only variables present, e.g. A and B.Please help needed! Reply Link to post Share on other sites More sharing options...
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