CHAPTER 9 SYSTEM OF MATRICES.pdf - CHAPTER 9 SYSTEMS OF EQUATIONS MATRICES AND DETERMINANTS L35 DEFINE THE SYSTEM OF LINEAR EQUATIONS AND MATRICES

CHAPTER 9 SYSTEM OF MATRICES.pdf - CHAPTER 9 SYSTEMS OF...

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1 CHAPTER 9: SYSTEMS OF EQUATIONS, MATRICES AND DETERMINANTS L35 DEFINE THE SYSTEM OF LINEAR EQUATIONS AND MATRICES. DEFINE THE TYPES OF MATRICES AND ECHELON MATRICES FORM. 9.1 System of Linear Equations Consider a system of two linear equations in two variables given by ax by c f ey dx where a, b, c, d, e and f are real constants with a and b are not both zero and d and e are not both zero. The graph of each of the above equation is a straight line. The solution to the system is the point of intersection of the two lines. If 1 L and 2 L represent the first and second equation, respectively then i. 1 L and 2 L intersect at only one point, in which case the system has exactly one solution. ii. 1 L and 2 L parallel, in which case the system has no solution. iii. 1 L and 2 L coincide, in which case the system has infinitely many solutions. Defn 9.1.3 (Consistent / Inconsistent System) o System with solutions is called a consistent system. o System without solution is called an inconsistent system. Eg :
2 Determine whether the following SLE are consistent or not: a) 7321x76221xxEg : Find the constant ksuch that the following system of linear equations has infinite many solutions. Hence find the set of solutions to the system. kyxyx223We can extend our discussion to the system of three equations in three variables. Consider a system of three linear equations in three variables given by 1131211bzayaxa2232221bzayaxa3333231bzayaxawhere iijba,are real constants and the coefficients of zyx,,are not simultaneously zero. The solution to the system is the point of intersection of the three 1 2 1 2 x
Eg :
4 9.2 System of Linear Equations in Matrix Form - Augmented Matrices A system of equations 6 2 2 4 z y x 4 8 3 2 z y x 14 8 2 5 z y x can be written in the format of rectangular array of numbers 14 4 6 8 2 5 8 3 2 2 2 4 14 4 6 8 2 5 8 3 2 2 2 4 Augmented matrix. 8 2 5 8 3 2 2 2 4 Coefficient matrix. Equations Matrices Write the system 6 2 2 4 z y x 4 8 3 2 z y x 14 8 2 5 z y x System in Augmented Matrix form 14 4 6 8 2 5 8 3 2 2 2 4 Divide 1 st eq by 4 2 3 2 1 2 1 z y x 4 8 3 2 z y x 14 8 2 5 z y x 14 4 2 / 3 8 2 5 8 3 2 2 / 1 2 / 1 1
5 Multiply 1 st eq by 2 and add to 2 nd eq 2 3 2 1 2 1 z y x 7 9 2 z y 14 8 2 5 z y x 14 7 2 / 3 8 2 5 9 2 0 2 / 1 2 / 1 1 Multiply 1 st

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