 # Consider a system of three linear equations in three variables. Give examples of two reduced forms that are not row-equivalent if the system is: a) onsistent and dependent. b) Inconsistent djeljenike 2020-10-26 Answered
Consider a system of three linear equations in three variables. Give examples of two reduced forms that are not row-equivalent if the system is: a) onsistent and dependent. b) Inconsistent
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Step 1 Answer for (a). Let us consider following two systems

Notice that augmented systems of the above two systems are as follows

Note that the reduced forms the above two systems are

Therefore reduced forms of the systems (I) and (II) are not row-equivalent. Notice that both systems are consistent and dependent because solutions to the systems are

Step 2 Answer for (b). Let us consider following two systems

Notice that augmented systems of the above two systems are as follows

Note that the reduced forms the above two systems are Therefore reduced forms of the systems (I) and (II) are not row-equivalent. Further note that both the systems are inconsistent because from last row of reduced from we see that $0=1$, which is not possible.