# 1.5 - Section 1.5 Reduced Echelon Form 1 0 r If we get 0 1...

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Section 1.5 Reduced Echelon Form If we get 10 01 r s then the solution is x 1 = r and x 2 = s. If we get 100 010 001 r s t    then the solution is x 1 = r, x 2 = s and x 3 = t. With many systems, particularly rectangular systems, this isn't possible. We need a form to show us when the augmented matrix is "as reduced as possible".

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Reduced Echelon Form All four of these conditions must be met: (1) All rows that consist entirely of zeros are at the bottom. (2) In a nonzero row, the first nonzero entry from the left is a 1. This is called a "leading one". (3) The leading ones form a stairstep pattern down and right. (4) The leading one in each row is the only nonzero element in its column.
Examples: Are the following matrices in Reduced Echelon form? (A) 1023 0154 0000    YES (B) 0110 0001 YES (C) 1002 0151 N O (row of 0’s isn’t at bottom) (D) 11 0 2 0100 005 1 0 NO (1 st nonzero entry in 3 rd row isn’t a 1)

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(E) 001 100 010    N O (Leading 1’s don’t form a stairstep pattern down and right) (F) 170 1 0 40
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1.5 - Section 1.5 Reduced Echelon Form 1 0 r If we get 0 1...

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