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**Unformatted text preview: **Math 313 Lecture #2 § 1.2: Row Echelon Form Not Reaching Strict Triangular Form. The algorithm that reduces the augmented matrix of an n × n system to that of an equivalent strictly triangular system fails when at any stage of the reduction the only choice for a pivot element is zero. Example. Find, if possible, the augmented matrix of a strictly triangular system that is equivalent to x 1 + x 2 + 2 x 3- x 4 = 1 ,- 2 x 1- 2 x 2- 3 x 3 + 5 x 4 = 0 , x 1 + x 2 + 3 x 3 + 2 x 4 = 3 , x 3 + 3 x 4 = 6 . Apply the reduction algorithm to the augmented matrix for this system: 1 1 2- 1 | 1- 2- 2- 3 5 | 1 1 3 2 | 3 1 3 | 6 R 2 + 2 R 1 → R 2 R 3- R 1 → R 3 ⇔ 1 1 2- 1 | 1 0 0 1 3 | 2 0 0 1 3 | 2 0 0 1 3 | 6 no nonzero pivot element in second column below first row R 3- R 2 → R 3 R 4- R 2 → R 4 ⇔ 1 1 2- 1 | 1 0 0 1 3 | 2 0 0 0 | 0 0 0 | 4 (1 / 4) R 4 → R 4 followed by R 3 ↔ R 4 ⇔ 1 1 2- 1 | 1 0 0 1 3 | 2 0 0 0 | 1 0 0 0 | Reinserting the variables in the last row of this augmented matrix gives x 1 + 0 x 2 + 0 x 3 + 0 x 4 = 0 which is always true. Reinserting the variables in the third row gives x 1 + 0 x 2 + 0 x 3 + 0 x 4 = 1 which is never true. Thus the system corresponding to the last augmented matrix is inconsistent. Since the elementary row operations preserve the solution set (they give equivalent sys- tems), the original system is inconsistent as well. / / / / Example. Find, if possible, the augmented matrix of a strictly triangular system that is equivalent to x 1 + x 2 + 2 x 3- x 4 = 1 ,- 2 x 1- 2 x 2- 3 x 3 + 5 x 4 = 0 , x 1 + x 2 + 3 x 3 + 2 x 4 = 3 , x 3 + 3 x 4 = 2 ....

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