To determine Absence of a unique solution to the system which has more variables than equations. Explanation Given: For a system of equations to be consistent rank of augmented matrix should be equal to coefficient matrix of system. Here exist two conditions: 1. The rank of augmented matrix is equal to the rank of coefficient matrix of system and is equal to the number of equations and variables. Then, there exists a unique solution to the system. By fundamental rules of algebra, it is definitive that x number of equations is required to determine x number of variables. 2. The rank of augmented matrix is equal to the rank of the coefficient matrix of system and is less then to the number of equations in the system. Then, the system is considered as consistent but there exists an infinite number of solutions. In such case, if there are 3 variables ( x , y , z ) in a system, there is always a variable (let z ): ∀ x , y ; z ∈ [ − ∞ , ∞ ] . Explanation: Consider a system of equations as mentioned below: 4 x − y = 2 − 12 x + 3 y = − 6 The system has two variables, therefore at least two equations are required to determine the solution. Write the set of equations in form of augmented matrix and perform row operations to reduce the matrix to RREF . A = [ 4 − 1 − 12 3 ] , B = [ 2 − 6 ] And, [ A : B

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Author: Lial

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Chapter 6.2, Problem 72E

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Absence of a unique solution to the system which has more variables than equations.

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Chapter 6.2, Problem 71E

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