Matrix Binomials

[ibastudent]

Hi, what do they mean by "Use an algebraic method to explain how you arrived at your general statement"???

[Pod]

Well, I think he means a formal justification. Instead of an informal one, where you just test different values, he is asking you to algebraically get your general statement. This is called a formal justification. Hope that helps.

[herong2x]

It will be easier to write and explain your paper if you work it out yourself. I did the same investigation, maybe i can help you. What are you stuck with?

i've only started the first part of this investigation and i was wondering, how can we improve on the analysis of this investigation (like the part of the rubic

the communication part and also the use of technology.) i know that for use of technology, we can use graphs, diagrams or any tables etc to enhance the portfolio but i just cant see how

my portfolio is due in two weeks i dont want to cram it like the first one (i did on the logarithms and got an 8/20 ) im aiming for at least a 14-16

not higher than that for now hehe

Thanks for your help! I just turned it in. Hope I did well x]

may i ask what you got for this portfolio?

Im desperate right now. Im handing in my portfolio tomorrow and Im not even half way. Can anyone help?

could i ask what you got in this portfolio?

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!

could i ask you how you got this?

[Cubz]

Ok so im almost finished this one,

I had an idea for the (A+B)n which most of you seem to have a lot of trouble with (i did to)

You'll notice that matrix (A+B) is the same as matrix M when simplified so that therefore your general statement should be the same as the expression for (A+B)n. You can go about finding the expression by doing different values for n and then expanding the matrix with a and b. You'll find that you come out with 2a^2 + 2b^2 etc. for (A+B)^2 etc. i don't think i can be more explicit than that but hope you get my drift......

This was actually how i showed that M= A+B initially but i just moved it up when i realised you could use it for (A+B)n - TOOK ME WAYYYYYYY TOO LONG lol

Anyway i am having real trouble with the algebraic justification at this point, a few people have just algebraically reiterated how M^n = A^n + B^n but i don't think thats it, they do seem to be asking for more. But yeah, seeing as we never covered the formal proof in our class our teacher told us we weren't required to do that, but he wasn't able to be more specific as to what we were supposed to do. We also haven't done induction or anything else even resembling proofs. So am completely stumped.

Another question regarding negative indices. People in my class are saying that because Matrix X can't be to a negative indice (as the determinant is undefined) the general statement can be to a negative indice. I don't agree with this as it the negative indices do work for matrix M. Any suggestions?

[Mayo]

I'm working on my second mathematics portfolio (Type I SL) on Matrix Binomials, and I've done most of the work, except there's one part that I just can't seem to work out, and it's really really bugging me!

Here's the first part of the portfolio:

Let X=(1,1,1,1) and Y=(1,-1,-1,1) (2x2 matrices)

Let A=aX and B=bY, where a and b are constants.

Use different values of a and b to calculate A², A³, A⁴; B², B³, B⁴

By considering integer powers of A and B, find expressions for Aⁿ, Bⁿ, (A+B)ⁿ

I've already done all of that, except for the expression for (A+B)ⁿ.

I keep thinking of the binomial theorem when I see it, but I can't seem to apply it to the matrices.

Any ideas would be really appreciated!

[deissi]

A good idea is to start by looking at old topics with the same exact title as yours, instead of starting a completely new topic.

[Cubz]

I'm working on my second mathematics portfolio (Type I SL) on Matrix Binomials, and I've done most of the work, except there's one part that I just can't seem to work out, and it's really really bugging me!

Here's the first part of the portfolio:

Let X=(1,1,1,1) and Y=(1,-1,-1,1) (2x2 matrices)

Let A=aX and B=bY, where a and b are constants.

Use different values of a and b to calculate A², A³, A⁴; B², B³, B⁴

By considering integer powers of A and B, find expressions for Aⁿ, Bⁿ, (A+B)ⁿ

I've already done all of that, except for the expression for (A+B)ⁿ.

I keep thinking of the binomial theorem when I see it, but I can't seem to apply it to the matrices.

Any ideas would be really appreciated!

Ok so apparently there are a number of different ways to find an expression for (A+B)n, one involves the binomial theorem but i personally do not get this at all. My suggestion is to look at doing it algebraically as i explained above, as it is wayy simpler. If you don't understand how i explained it i can have a go at it again as it did ramble a bit my bad.

Anyway you said you have done it all except for that. How did you do the algebraic method. I have spoken to ppl who studied maths at uni and can't seem to get a clear answer on this !!!

Freaking out here

[sweetnsimple786]

If I remember this assignment correctly, then you just need to realize this:

A scalar is simply a number--1, 4. 1/3, whatever

If you take the identity matrix and multiply it by a certain scalar, you may find patterns that relate to A and B. If you don't then you're not at that part yet.

[Mayo]

Ok so apparently there are a number of different ways to find an expression for (A+B)n, one involves the binomial theorem but i personally do not get this at all. My suggestion is to look at doing it algebraically as i explained above, as it is wayy simpler. If you don't understand how i explained it i can have a go at it again as it did ramble a bit my bad.

Anyway you said you have done it all except for that. How did you do the algebraic method. I have spoken to ppl who studied maths at uni and can't seem to get a clear answer on this !!!

Freaking out here

Haha, I'm freaking out too! But I'm getting the hang of it, luckily, I have a week left to complete the portfolio. Thanks for your help, I understand how to get (A+B)ⁿ, and binomial was not a good way to go for me!

When they say algebraic method, I think they mean clearly justify the general statement in a different way, because we have already justified it through the testing of different values, but I'm actually working it this at the moment but I'm still unsure on what they specifically expect from us. I think I'm rambling.

But I'll just keep working on it, hopefully I'll have a eureka moment sometime soon?

[Cubz]

Ok so apparently there are a number of different ways to find an expression for (A+B)n, one involves the binomial theorem but i personally do not get this at all. My suggestion is to look at doing it algebraically as i explained above, as it is wayy simpler. If you don't understand how i explained it i can have a go at it again as it did ramble a bit my bad.

Anyway you said you have done it all except for that. How did you do the algebraic method. I have spoken to ppl who studied maths at uni and can't seem to get a clear answer on this !!!

Freaking out here

Haha, I'm freaking out too! But I'm getting the hang of it, luckily, I have a week left to complete the portfolio. Thanks for your help, I understand how to get (A+B)ⁿ, and binomial was not a good way to go for me!

When they say algebraic method, I think they mean clearly justify the general statement in a different way, because we have already justified it through the testing of different values, but I'm actually working it this at the moment but I'm still unsure on what they specifically expect from us. I think I'm rambling.

But I'll just keep working on it, hopefully I'll have a eureka moment sometime soon?

Awesome glad you get it. I just clarified it with my teacher. there are several ways to prove M= A+B so just do it two different ways.

Ie. you can do it M=A+B and then M^2=A^2 + AB + BA + B^2 you'll notice that AB and BA equals the zero matrix and go from there, establish a pattern and so on.

hope that helps.

GOOD LUCK!!!!!

[IBerrr]

hi i'm also doing this and i'm not clear on what they mean by "use an algebraic method to explain how you arrived at your general statement". i have my general statement, but i'm stuck on the algebraic method

[Sandwich]

hi i'm also doing this and i'm not clear on what they mean by "use an algebraic method to explain how you arrived at your general statement". i have my general statement, but i'm stuck on the algebraic method

Have you tried substituting algebraic values into your matrix and then working that matrix out by hand?

[fatalmatrix]

HELP! My portfolio is due in three days!

First, I don't know the format of the whole portfolio because my teacher didn't explain it AT ALL. We didn't even get a rubric. There should be an intro and a conclusion right? Should the paper be double spaced? Should there be a title page? What's on the title page? How much should we explain? I mean, do we assume that the readers are already familiar with adding and multiplying matrices or do we have to explain every step from the beginning?

Also, aren't there two general statements? Which one do they want? It says to find M^n in terms of aX and bY, isn't that kind of obvious? Since M = aX + bY?

[Mahuta ♥]

First, I don't know the format of the whole portfolio because my teacher didn't explain it AT ALL. We didn't even get a rubric. There should be an intro and a conclusion right? Should the paper be double spaced? Should there be a title page? What's on the title page? How much should we explain? I mean, do we assume that the readers are already familiar with adding and multiplying matrices or do we have to explain every step from the beginning?

Introduction is just about saying what you have to do and what you're trying to find, which is basically what is written on the paper given to you by the teacher. The conclusion is summing up your findings and basically making an overall conclusion related to what you were asked to investigate.

Also, aren't there two general statements? Which one do they want? It says to find M^n in terms of aX and bY, isn't that kind of obvious? Since M = aX + bY?

There is more to it than you what you see, you have to look into it more considering your previous findings, consider both algebraic ways and the normal ways.

[sweetnsimple786]

HELP! My portfolio is due in three days!

First, I don't know the format of the whole portfolio because my teacher didn't explain it AT ALL. We didn't even get a rubric. There should be an intro and a conclusion right? Should the paper be double spaced? Should there be a title page? What's on the title page? How much should we explain? I mean, do we assume that the readers are already familiar with adding and multiplying matrices or do we have to explain every step from the beginning?

Also, aren't there two general statements? Which one do they want? It says to find M^n in terms of aX and bY, isn't that kind of obvious? Since M = aX + bY?

I didn't do an intro or conclusion. My title page had the following: title [Matrix Binomial], name, type of project [sL Type 1], and date [02 April 2009].

Then I just started "talking." You will want to explain how to multiply matrices. The easiest way is to use variables. Also, if in the course of your paper, you do some fancy things with matrices, you should explain how to do them either in the beginning or along the way. Your choice.

Mine is not doublespaced, but that is because I used MathType (an awesome program that makes doing the matrices a lot easier) for the majority of my paper. You can double or 1.5 space it.

I got an 18/20 on mine because I didn't explain the algebraic method at the end well enough. I'd upload mine, but all of my papers are in limbo because I'm not graduating for another year.

Edit: Oh yeah, and about your statements. Our teacher told us there are at least two ways to get a pattern or statement, but she said that one way is easier than the rest. If you're stuck when you get to the M, maybe you want to consider messing with your previous general statements, using or removing scalars or something?

For the M^n, it's not hard--just do some substituting from previous statements. So if you think it's too easy to be right, it probably isn't. Just plug in numbers and stuff

Edited May 29, 2009 by sweetnsimple786

[fatalmatrix]

Thank you so much for helping me with the format and the general statement, I think I get it now. But I've ran into a new problem: I can explain (A+B)^n perfectly using variables, but I can't do it with integer values. Will I get a worse mark if I didn't use "different numerical values of a and b" to arrive at my expression for (A+B)^n?

When you say

There is more to it than you what you see, you have to look into it more considering your previous findings, consider both algebraic ways and the normal ways.

What do you mean by the "algebraic ways" and the "normal ways"? Is one with variable and the other with numerical values or am I completely missing the point here?

[Mahuta ♥]

Will I get a worse mark if I didn't use "different numerical values of a and b" to arrive at my exp​ression for (A+B)^n?

You probably will yeah, so i think you should do it.

The thing is, I dont remember this one very well but:

Algebraic way is basically not using matrices, just using 'a' and 'b', whereas the normal way is using matrices.

[fatalmatrix]

What? How do you use numbers to arrive at the expression for (A + B)^n?

It is OK if I don't write the portfolio in the same order as the instruction steps? I find it very odd to leave the algebraic method 'till the very end.

[fatalmatrix]

Can you use matrix notation in an "expression"?

For example, is Y = (e^n, 0, 0, e^n) an "expression"?

Or do you have to put it as Y = e^n I?

[narumi]

can someone please explain to me what a maths portfolio is?! i got the booklet already, but don't know what to do about it. should i just answer all the questions given or what? i am really on a blank stage right now. my teacher doesn't do much explanation on portfolios.

could someone please help me!!!