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, which utilizes things called matrices. Defnition 15.1. If m and n are positive integers then an m × n matrix (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 dimension of the matrix is m × n where the matrix has m rows (horizontal) and n columns (vertical). Notation: If A is an m × n matrix we will sometimes write A =( a ij ) so that we may refer to the (i,j)-entry of A as a ij . Example 1. A a ij )= 12 34 56 is a 3 × 2 matrix with a 12 =2 ,and a 32 =6. Defnition 15.2. A common use of matrices ( that we will be studying here) is to represent a system of linear equations. The matrix derived from the coefficients and the constant terms is called the augmented matrix of the sys- tem, and the matrix derived from only the coefficients is called the coefficient matrix of the system. 1
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Example 2. We give a system and its corresponding augmented and coeffi- cient matrices. x 4 y +3 z =5 x +3 y z = 3 2 x 4 z =6 System 1 43 5 13 1 3 20 4 6 Augmented Matrix 1 1 4 Coefficient Matrix Defnition 15.3. Looking at the example above we can see that it is easy to translate a system to it’s augmented matrix, and to translate from the aug- mented matrix back to the system. So manipulating a system of equations ( using the three elementary operations discussed in the previous section) in order to solve it is equivalent to somehow manipulating the augmented matrix to solve the system. The corresponding operations on the augmented matrix are called the elementary row operations ,( ERO ’s). We now show how each elementary operation translates to an elementary row operation: Elementary Operation Elementary Row Operation (1) Multiply any equation by (i) Multiply any row by a non-zero scalar a non-zero scalar (2) Interchange the position (ii) Interchange the position of two equations of two rows (3) Replace any equation with (iii)Replace any row with the sum of itself and a multiple the sum of itself and a multiple of another equation from the system of another row from the matrix Theorem 15.4. Applying any sequence of elementary row operations to the augmented matrix of a system does not change the solution(s) of the under- lying system. Defnition 15.5. We say that two matrices are row equivalent if one matrix can be transformed into the other by applying a series of elementary row operations. 2
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Example 3. For example the matrices: (a) 1 43 13 1 20 4 and 28 3 1 4 are row equivalent since multiplying row 1 of the ±rst matrix by (-2) (written R 1 →− 2 R 1 ) yields the second matrix.
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UNIT14 - Unit 14 Matrices and Systems of Linear Equations...

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