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matix bimonals

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Introduction

Mathematics Portfolio – Matrix Binomials                                                                                            Samed Nakhla

Matrix Binomials

Question

  1. Let image00.pngimage00.png

Solution

image09.png

image17.png

image25.png

image33.png

image42.png

image48.png

Question

  1. By considering integer powers of X and Y, find expressions for image54.pngimage54.png

Following the pattern in the first question and noticing from it, I deduce that the formula for:

image01.png

image02.png

Also to find the image03.pngimage03.png power of (X+Y) I will use some examples to find the pattern in it:

image04.png

image05.png

image06.png

From the examples above, I deduced the equation of image07.pngimage07.png by following the pattern, and it is:

image08.png

Question

Let A=aX and B=bY, where a and b

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Middle

a and b

Let’s assume that a= 2

image13.png

image14.png

image15.png

Also, let’s assume that b=3

image16.png

image18.png

image19.png

To obtain the general formula for image20.pngimage20.png we need to find a formula for image10.pngimage10.pngwithout using different values of a and b, are:

image21.png

image22.png

image23.png

image24.png

image26.png

image27.png

Question

By considering integer powers of A and B, find expressions for image20.pngimage20.png

Solution

By following the pattern in the previous question, one can deduce that the formulas are:

image28.png

image29.png

And to find the formula forimage30.pngimage30.png, one should give some examples

...read more.

Conclusion

bY.

Solution

After following the pattern in the previous question, I concluded the formula ofimage46.pngimage46.png to be:

image47.png

To obtain the general formula ofimage46.pngimage46.png in terms of aX and bY:

image49.png

Question

Test the validity of your general statement by using different values of a, b and n.

Solution

Let’s assume that a=2, b=3 and n=3

image50.png

And by using the formulas we got from before image51.pngimage51.png

Where image52.pngimage52.png

So, also by assuming the same variables as before a=2, b=3 and n=3, and then replacing them in the equationimage51.pngimage51.png, we get the exact result as before:

image53.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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