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# lecture_4 - MA 35100 LECTURE NOTES WEDNESDAY JANUARY 20...

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MA 35100 LECTURE NOTES: WEDNESDAY, JANUARY 20 Gauss-Jordan Elimination We review some ideas from the last lecture. First, we begin with a formal deﬁnition. Deﬁnition. A matrix is in reduced row-echelon form (rref) if it satisﬁes all of the following conditions: a. If a row has nonzero entries, then the ﬁrst nonzero entry is 1, called the pivot in this row. The variables corresponding to the pivots are called leading variables , and the other variables are called free variables . b. If a column contains a pivot, then all other entries in that column are zero. c. If a row contains a pivot, then each row above contains a pivot further to the left. Recall that in the previous lecture we began with the matrix A = 0 0 1 - 1 - 1 4 2 4 2 4 2 4 2 4 3 3 3 4 3 6 6 3 6 6 0 0 0 0 0 0 . and performed a series of steps to arrive at the matrix E = 1 ± 2 0 3 0 2 0 0 1 ± - 1 0 2 0 0 0 0 1 ± - 2 0 0 0 0 0 0 0 0 0 0 0 0 . Note that A is not in reduced row-echelon form, but E is. The pivots are circled in the matrix above. Given a matrix

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lecture_4 - MA 35100 LECTURE NOTES WEDNESDAY JANUARY 20...

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