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SECTION 1.2 ROW REDUCTION AND ECHELON FORMS Nonzero row or column Leading entry Echelon form Reduced echelon form FACT: Every matrix is row equivalent to a unique reduced echelon form matrix. Pivot, pivot position and pivot column ROW REDUCTION ALGORITHM: 1. Begin with the leftmost nonzero column. This is a pivot column and the pivot position is at the top. 2. Select a nonzero entry in the pivot column and move its row to the top. 3. Create zeros in all positions below the pivot. 4. Ignoring the row containing the pivot position and all rows, if any, above it. Repeat Steps 1 to 3 for the remaining submatrix. Continue until there are no more nonzero

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Unformatted text preview: rows. 5. Start with the rightmost pivot, work upward and to the left, and make zeros above each pivot. Make each pivot a 1 by dividing its row by its value when necessary. Find the general solutions of the systems whose augmented matrices are given. Indicate the pivot positions and the pivot columns. 1 0 3 2 0-8 0 0 1 0 2 3 0 0 0 0 1 1 0 0 0 0 0 1 3 5 2 2 8 18 10 3 13 31 18 Suppose a 3 × 5 coeﬃcient matrix for a system has three pivot columns. Is the system consistent? Why or why not? HOMEWORK: SECTION 1.2, # 4, 8, 14, 16, 20, 24, 25, 26...
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