# matrix power

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. In mathematics, a matrix is a rectangular table of elements, or entries, which may be numbers or, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra. The purpose of this mathematical paper is to explore and investigate the powers of matrix.

A matrix may be squared or even raised to an integer power. However, for a matrix to be able to be raised to the power the matrix must be a square matrix, meaning that it must be orthogonal in shape and contains the same width and height all around.

The following list describes the effects on a matrix when a matrix is raised to an integer power:

Raising a matrix to a power greater than one involves multiplying the matrix by itself a specific number of times.

For example, if, and

In order to solve the matrix power above, we multiply the matrix by “n” number of times.

Thus, if, and, then to solve we times the matrix “M” by two times.

To multiply matrices, we take the rows of the left hand matrix and pair it with the column of the right hand matrix. The first step is to take the first row of the left hand matrix, in this case being [2 0], write it as a column and then pair it up with the first column of the right hand ...