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matrix power

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Introduction

International Baccalaureate

Mathematics Portfolio - Standard Level Type I

Matrix Powers

Student Names: Nam Vu Nguyen

Set Date: Wednesday, February 13, 2008

Due Date: Tuesday, March 04, 2008

School Name: Father Lacombe Senior High School

Teacher: Mrs. Stephanie Gabel

I CERTIFY THAT THIS PORTFOLIO ASSIGNMENT IS ENTIRELY MY OWN WORK

                                Nam Vu Nguyen: ____________________________________

International Baccalaureate

Mathematics Portfolio - Standard Level - Type I

Matrix Powers

Mathematics is the science where the concepts of quantity, structure, space and change is studied. A science where patterns are discovered in numbers, in space, in science, computers, imaginary abstractions, and everything else contained in the universe. It is the type of science that draws conclusions and connections to the world’s analytical problems that exists all around us. Mathematics is used to describe the numerous natural phenomena that occur around us every day. Today math is being applied and developed into numerous evolving educational fields, inspiring humans to discover and make use of their mathematical knowledge, which will in turn lead to entirely new discoveries. An example of the usage of mathematics in society today would be the use and manipulation of matrices in the field of computer graphics.

The Matrix theory is a branch of mathematics that focuses on the study of matrices. Originally it is a sub-branch of linear algebra, yet it has grown to cover subjects related to graph theory, algebra, combinatory, and statistics as well.

...read more.

Middle

image10.png

Based on the following data above we concluded that there are two sets of methods to solve the powers of matrices. There is also another topic that should also be brought light upon. There exists a pattern in the end results of the powers of the matrices. The final matrices themselves form an identity matrix where the final non-zero digit is the power of the original non-zero digit, and the zeros remain the same and unchanged. In linear algebra, the identity matrix of size “n” is a n-by-n square matrix with either ones or non-zero digits on the main diagonal and zeros elsewhere. Sometimes the zeros could be replaced by a different number. In this case the power of the non-zero digit, being 2 in this report, replaces the non-zero digit in the identity matrix, and the zeros remain the same. Therefore the general expression of a Mn matrix in terms of “n is simply:

image11.png

When given the matrices image13.pngandimage14.png, we will now test and analyze the pattern observe from before and also modify our pattern conclusion and our methods of solving for the powers of a matrix.

First we shall calculate image15.pngby using a Ti-83 Graphing calculator. To enter a matrix into a Ti-83 Graphing calculator we would need to follow the following steps.

In order to enter the matrix into the Ti-83, press the [MATRIX] key, select [EDIT]  and then [MATRIX A].

...read more.

Conclusion

The Matrix theory is a branch of mathematics that focuses on the study of matrices. A matrix is a rectangular table of elements, or entries, which may be numbers or, any abstract quantities that can be added and multiplied. A matrix may be squared or even raised to an integer power. However, for a matrix to be able to be raised to a power the matrix must first be a square matrix. Different values of a power whether it be an integer or a whole number, can create and manipulate the original matrix into a different form or branch of matrices. The portfolio not only investigates the powers of matrix but it also investigates the patterns and the meaning and value of everything beneath the surface. Matrices are created by humankind to further their technological advances and create a better society and increase the level of living standards in the world. The importance of understanding matrices and their operations will help students and even those who no longer participate in school, will help them create a better should they advance forward into the technological and engineering fields of careers. The purpose of this portfolio was to develop a skill necessary in that area of career, a skill of being able to decipher patterns and know how to manipulate them in order to apply them to the real world.

...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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