Systems of linear equations

Systems of linear equations - Systems of linear equations...

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Systems of linear equations Advanced Level Pure Mathematics Advanced Level Pure Mathematics Algebra Chapter 9 Systems of Linear Equations 9.1 Introduction and Existence and Uniqueness of Solution 2 9.3 Gaussian Elimination 7 9.4 Solutions of Systems of Linear Equations 8 9.5 Solution of a Homogeneous System of Equations 13 9.1 Introduction and Existence and Uniqueness of Solution Prepared by K. F. Ngai Page 1 9

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Systems of linear equations Advanced Level Pure Mathematics Consider three equations in three unknowns, i.e. = + + = + + = + + 3 3 33 2 32 1 31 2 3 23 2 22 1 21 1 3 13 2 12 1 11 b x a x a x a b x a x a x a b x a x a x a i.e In Matrix From. = 3 2 1 3 2 1 33 32 31 23 22 21 13 12 11 b b b x x x a a a a a a a a a The system of three linear equations may be rewritten as B AX = If 0 B , the system is called a non-homogeneous system of linear equations. For 0 A , 1 A - exist. The system becomes B AX = B A X 1 - = Three linear equations will have a unique solution. In this case, three linear equations are said to be linearly independent. Theorem 9-1 Let A be a square matrix. If A is a non-singular matrix, i.e. 0 A det , then the system of linear equations B AX = has a unique solution given by B A X 1 - = . Example 1 Use the method of inverse matrix to solve the system of equations = + = - 3 y x 4 y 2 x 3 Example 2 Solve the system of equations Prepared by K. F. Ngai Page 2
Advanced Level Pure Mathematics - = + + - = - - = - + 3 z 7 y x 4 11 z y 7 x 4 4 z y x 2 Example 3 Solve = + + + = + - - = - - + 0 3 7 4 0 11 7 4 0 4 2 z y x z y x z y x Example 4 Discuss the number of solutions to the following systems of linear equations: (a) = + = + 4 4 2 3 2 y x y x (b) = + = + 6 4 2 3 2 y x y x Theorem 9.2 Let A be a n n × matrix. If 0 det = A , then the linear system b Ax = has no solution or infinitely many solutions. Prepared by

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This note was uploaded on 11/26/2011 for the course COMPUTER S 1003 taught by Professor Angelosstavrou during the Spring '11 term at King Saud University.

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Systems of linear equations - Systems of linear equations...

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