lecture6 - means interchanging i th and j th rows. 2. A row...

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Math 006 (Lecture 6) Augmented Matrices Example 1. Solve the system of linear equations ± 3 x 1 + 4 x 2 = 1 x 1 - 2 x 2 = 7 Method of Elimination Augmented Matrix Definition 1. A matrix is a rectangular array of numbers written within brackets. Example 2. A = ² 1 - 5 7 0 1 / 2 - 1 / 3 ³ , B = 3 7 6 . 8 2 . 3 1 / 2 2 . Each number in a matrix is called an element of the matrix. If a matrix has m rows and n columns, it is called an m × n matrix . A matrix with n rows and n columns is called a square matrix of order n . The position of an element in a matrix is given by the row and the column containing the element. This is denoted by a ij . The principal diagonal of a matrix A consists of the element a 11 , a 22 , a 33 , . . . . 1
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In general, associated with each linear system of the form ± a 11 x 1 + a 12 x 2 = k 1 a 21 x 1 + a 22 x 2 = k 2 where x 1 and x 2 are variables, is the augmented matrix of the system: ² a 11 a 12 ³ ³ k 1 a 21 a 22 ³ ³ k 2 ´ . We solve the linear system by transforming the augmented matrix using a combination of the following operations: 1. Two rows are interchanged: R i R j
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Unformatted text preview: means interchanging i th and j th rows. 2. A row is multiplied by a nonzero constant: R i → kR i means multiplying i th row by a nonzero constant k . 3. A constant multiple of one row is added to another row: R i → R i + kR j means adding i th row by k multiple of j th row. Notice that the above operations are called the row operations . Target: Reduced the augmented matrix to the forms (A) ² 1 0 ³ ³ m 0 1 ³ ³ n ´ , (B) ² 1 m ³ ³ n ³ ³ ´ and (C) ² 1 m ³ ³ n ³ ³ p ´ . Conclusion: 1. We have a unique solution for our system if it reduces to Form A. 2. We have infinitely many solutions if it reduces to Form B. 3. We have no solution if it reduces to Form C. 2 Example 3. Solve the following systems of linear equations: (a) ± 2 x 1-x 2 = 4-6 x 1 + 3 x 2 =-12 (b) ± 2 x 1 + 6 x 2 =-3 x 1 + 3 x 2 = 2 3...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture6 - means interchanging i th and j th rows. 2. A row...

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