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Unit14 - Unit 14 Matrices and Systems of Linear Equations...

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Unit 14 Matrices and Systems of Linear Equations In this section we will learn a method known as row-reduction for solving SLE’s. This method works in basically the same way as the method of elim- ination, but utilizes a structure called a matrix to eliminate the repetitive writing down of symbols which never change from one step to the next. First, we must define what this mathematical structure called a matrix is. Definition 14.1. If m and n are positive integers then an m × n matrix ( note: m × n is read “ m by n ”) is a rectangular array of numbers: a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 a m 3 · · · a mn in which each number a ij is called the ( i, j ) -component or ( i, j ) -entry of the matrix. The descriptor m × n is called the dimension of the matrix, where m is the number of rows (horizontal) and n is the number of columns (vertical) in the matrix. Notation: Given a set of numbers a ij for i = 1 , ..., m and j = 1 , ..., n , we will sometimes use A = ( a ij ) to denote that A is the m × n matrix whose ( i, j )-entry is a ij . Also, when we have a matrix called A , we use a ij to denote the ( i, j )-entry of matrix A . That is, we use upper case letters to name matrices, and we use the lower-case version of the same letter, subscripted by row number (first subscript) and column number, to refer to an entry in the matrix. 1
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Example 1 . For A = ( a ij ) = 1 2 3 4 5 6 , determine the dimension of A and find a 12 and a 21 . Solution: Since A has 3 rows and 2 columns, A is a 3 × 2 matrix. Looking in row 1, column 2, we find that a 12 = 2. Similarly, looking in row 2 and column 1, we see that a 21 = 3. A common use of matrices that we will be studying here is to represent a system of linear equations. Definition 14.2. Consider any system of m linear equations in n variables, written in standard form. Let a ij be the coefficient, in the i th equation, of the j th variable. The m × n matrix A = ( a ij ) is called the coefficient matrix of the SLE. The augmented matrix for a SLE is obtained by appending the m right hand side values of the equations to the coefficient matrix, as an extra column in the matrix. We always delineate this extra column in an augmented matrix by placing a vertical line before it. Example 2 . Write the coefficient matrix and the augmented matrix for the following SLE: x - 4 y + 3 z = 5 - x + 3 y - z = - 3 2 x - 4 z = 6 SLE in standard form Solution: Since the system is already in standard form, we can obtain the coefficient matrix by simply writing the coefficients from the system in the same order in which they appear in the SLE. For instance, we get the first row of the coefficient matrix by extracting the coefficients of x , y and z , i.e. 1, - 4 and 3, from the first equation. Remember: Our definition of the coefficient matrix and the augmented matrix require that the system be in standard form.
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