Use the following properties to help justify/prove your answers to parts a, b an
ID: 3113512 • Letter: U
Question
Use the following properties to help justify/prove your answers to parts a, b and c. Prop 1: For a system with coefficient matrix A elementof R^mxn, the number of pivots is equal to the number of basic variables. Prop 2: For a consistent system with coefficient matrix A elementof R^mxn, there exists infinitely many solutions if and only if there is at least 1 free variable. Prop 3: A system with augmented matrix A elementof R^mxn in RREF is inconsistent if and only if there is a row with a all entries zero except for the entry in the augmented column. (a) For a system with coefficient matrix A elementof R^mxn, if the system has a unique solution then how many pivots must A have? (b) For a system with coefficient matrix A elementof R^mxn, if A has n pivots what is the largest m could be if we want to guarantee a unique solution? (c) For a system with coefficient matrix A elementof R^mxn, if the system has a unique solution which of the following relationships must be true? m nExplanation / Answer
(a). We know that a variable in a system of linear equations is called a basic variable if it corresponds to a pivot column. If not, the variable is known as a free variable. If the system has a unique solution, then there would not be any free variable(s).Further, if the system’s coefficient matix A Rmxn (i.e. the system has m equations in n variables), the number of pivots has to be n.
(b). For a consistent system with coefficient matix A Rmxn, if A has n pivots , then there is a solution if m n. There is a unique solution if m= n i.e. if the augmented matrix [ A|b ] has exactly n non-zero rows. The system will have infinite solutions if m < n ( as in such a case, there will be free variables).
(c). With a system with coefficient matix A Rmxn,if the system has a a unique solution, then we must have m = n.