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Unformatted text preview: Note that Y= Mat rix Binomials I ntroduction This paper will focus on the study of matrix binomials. In mathematics, a matrix ( plural matrices) is a rectangular array of numbers, referred to as ‘elements’, consisting of rows and columns. Matrices are similar to vectors but as mentioned before matrices are represented by blocks of number with rows and columns. The matrix illustrated above has 2 rows (horizontal components) and 2 columns (vertical components) and thus have and order of 2 × 2. In general, a matrix is used to store information, especially in the field of data management, and record data that depends on multiple parameters. Matrix multiplication connects matrices to linear transformations and has led to fractal geometry. When multiplying matrices the rows on the left hand matrix, the first matrix, have to pair up with the columns of the second matrix. For example, × 2 × 4 = 8 4 × 2 = 8 2 × 2 = 4 The first step is to take the first row of the first matrix [2 4 2] and pair it with the first column of the second matrix. Each pair is multiplied and then added together. The product achieved would be the element in the first row and first column of the answer. All combinations of rows and columns must be calculated. In case of square matrices, matrices with an equal number of rows and columns, more data can be extracted such as the determinant and inverse matrices. In relation to square matrices, matrix binomials refer to a situation consisting of two matrices of which both orders are 2 × 2. By applying the matrix binomials to different situations and testing the certain validities and claims general expressions and statements about the different matrix binomials would be reached. In this investigation one of the key focal points is to distinguish as well as relate different matrices in order to arrive at general statements and expressions concerning the matrix binomials. By manipulating matrix binomials through various calculations certain relations would be touched upon and related to the further calculations concerning the matrix binomials dealt with in the investigation. In this paper we will analyze the matrix binomials of X and Y and then its relations. The analysis of the matrices X and Y and all of the other operations will be carried out by algebraic methods. After calculating the matrix binomials of X and Y a general formula or statement would be reached. We will then focus on the matrices A and B, where A= a X and B= b Y. General statement would be reached for A n , B n and (A+B) n . Finally, the relationship on the matrices A and B will be investigated through the matrix M. We will then compare the different results achieved....
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- Spring '10
- Geometric progression, mat rix