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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Asma Vang · Accepted Answer

Step 1 Answer for (a). Let us consider following two systems  (I) {x − z=0y =0x + y − z=0 and (II) {x − y=0y − z=0x − z=0  Notice that augmented systems of the above two systems are as follows (I) [10−10010011−10] and (II) [1−10001−1010−10]  Note that the reduced forms the above two systems are  (I) [10−1001000000] and (II) [10−1001−100000]  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 Solution set of (I) : {(t, 0, t)| t ∈R} Solution set of (II) : {(t, t, t)| t ∈R}  Step 2 Answer for (b). Let us consider following two systems (I) {x − z=0y =0x + y − z=1 and (II) {x − y=0y − z=0x − z=1  Notice that augmented systems of the above two systems are as follows (I) [10−10010011−11] and (II) [1−10001−1010−11]  Note that the reduced forms the above two systems are (I) [10−1001000001] and (II) [10−1001−100001] 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.

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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