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Math Type I SL Matrices

Extracts from this document...

Introduction

Pure 20 IB Portfolio Project

Matrix Binomials

Dan Stoica

Henry Wise Wood High School

Calgary, Alberta

Mr. Bedford

Table of Contents

Title Page...........................................................................................................Page 1

Table of Contents............................................................................................Page 2

Mathematical Investigation Sheet...........................................................Page 3

Calculator/Introduction..............................................................................Page 4

Determining X and Y.....................................................................................Page 5

Determining X and Y continued...............................................................Page 6

Determining A and B.....................................................................................Page 7

Determining A and B continued...............................................................Page 8

Determining A and B continued...............................................................Page 9

Determining A and B continued............................................................Page 10

Determining A and B continued............................................................Page 11

Determining Expressions for (A + B)..................................................Page 12

Verifying (M).................................................................................................Page 13

Verifying (M) continued...........................................................................Page 14

Verifying (M) continued...........................................................................Page 15

Verifying (M) continued...........................................................................Page 16

Conclusion......................................................................................................Page 17

Mathematical Investigation Question

image00.jpg

Ti-84 Calculator Matrix Skills

How to enter a matrix

2nd  MATH -> Edit -> Choose “Letter” -> Enter Dimensions -> Enter Values

How to select a matrix

2nd MATH -> Select “Letter”

How to perform operations on matrices

2nd MATH -> Select Matrix -> Operation Key(+,-,*,/,) -> 2nd -> Select another Matrix -> Enter

How to put exponents on a matrix

2nd MATH -> Select Matrix -> Press ^ -> Press Number desired as exponent -> Enter

Introduction

        In the proceeding project, 2 by 2 matrices will be used to create a general statement. Different values will be used for constants and exponents.

...read more.

Middle

image17.png)2

= image18.pngimage18.png2

= image19.pngimage19.png

B3= (bY)3= (3image17.pngimage17.png)3

= image18.pngimage18.png3

= image20.pngimage20.png

B4= (bY)4= (3image17.pngimage17.png)4

= image18.pngimage18.png4

= image21.pngimage21.png

To test the above statement with a negative figure, Let “b” = (-3).

B2= (bY)2= (-3image17.pngimage17.png)2

= image22.pngimage22.png2

= image19.pngimage19.png

B3= (bY)3= (-3image17.pngimage17.png)3

= image22.pngimage22.png3

= image24.pngimage24.png

B4= (bY)4= (-3image17.pngimage17.png)4

= image22.pngimage22.png4

= image21.pngimage21.png

Therefore we can assume that the formula Bn=bn×Yn-1image17.pngimage17.png works in this case as well.

B2= b2xY2-1image17.pngimage17.png=

= 32ximage25.pngimage25.png = 9image26.pngimage26.png

= image19.pngimage19.png

B3= b3xY3-1image17.pngimage17.png=

= 33ximage28.pngimage28.png = 27image29.pngimage29.png

= image20.pngimage20.png

B4= b4xY4-1image17.pngimage17.png=

= 34ximage30.pngimage30.png = 81image31.pngimage31.png

= image21.pngimage21.png

B2= b2xY2-1image17.pngimage17.png=

= (-3)2ximage25.pngimage25.png = 9image26.pngimage26.png

= image19.pngimage19.png

B3= b3xY3-1image17.pngimage17.png=

= (-3)3ximage28.pngimage28.png =

(-27)image29.pngimage29.png= image24.pngimage24.png

B4= b4xY4-1image17.pngimage17.png=

= (-3)4ximage33.pngimage33.png = 81image26.pngimage26.png= image21.pngimage21.png

Decimals will be used for “b” in the following two examples.

B2=b2×Y2-1image34.pngimage34.png = (0.5)2×Y1image17.pngimage17.png

= 0.25×image25.pngimage25.png

= 0.25image26.pngimage26.png

= image35.pngimage35.png

B3=b3×Y3-1image34.pngimage34.png = (0.5)3×Y2image17.pngimage17.png

= 0.125image28.pngimage28.png

= 0.125image29.pngimage29.png

= image35.pngimage35.png

B4=b4×Y4-1image34.pngimage34.png = (0.5)4×Y3image17.pngimage17.png

= 0.0625×image30.pngimage30.png

= 0.0625image31.pngimage31.png

= image35.pngimage35.png

B2=b2×Y2-1image34.pngimage34.png = (0.25)2×Y1image17.pngimage17.png

= 0.0625×image25.pngimage25.png

= 0.0625image26.pngimage26.png

= image36.pngimage36.png

B3=b3×Y3-1image34.pngimage34.png = (0.25)3×Y2image17.pngimage17.png

= 0.015625×image28.pngimage28.png

= 0.015625image29.pngimage29.png

= image38.pngimage38.png

B4=b4×Y4-1image34.pngimage34.png = (0.25)4×Y3image17.pngimage17.png

= image39.pngimage39.png×image30.pngimage30.png

= image40.pngimage40.png

= image41.pngimage41.png

Therefore we can conclude that the formula Bn=bnYn works. From these examples, we can conclude that this statement works with positive, negative, and decimals for the “b” figure, just as it did for “a” when tested above. There are no limitations to this statement. Therefore, the statement is valid. A also check the answers using the matrix feature on the TI-84 calculator.

Patterns that were observed during this test were that whenever “

...read more.

Conclusion

Mn = anXn+bnYnis valid, and can be proved by using other expressions and by using examples, and provides answers that are consistent, that make sense, and that are logical.

Page

...read more.

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