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Suppose we were given the following matrix (I made up the numbers). The first and second row are in row echelon form. However the third is not. Do we still need to put the third row in row echelon form even if we can see that 1 * x = 2, which we could solve for x??

$$ \begin{pmatrix} 1&5&5\\ 0&1&9\\ 1&0&2\\ \end{pmatrix} $$

CuriousIndeed
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  • No, you do not (unless the problem is explicitly about transforming it to row echelon form). Row echelon form is useful for describing an algorithm that computes solutions of a general system, however you can come up with specific systems for which reducing to row echelon is superfluous. – Randy Marsh Oct 09 '20 at 13:36
  • But If we wanted to get the rank of the matrix the last transformation is needed or is there a trick? – CuriousIndeed Oct 09 '20 at 13:41
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    It is not always needed. You do not need to reduce $\pmatrix{0 & -71\ 11/128& 0}$ to row echelon form to figure out its rank. – Randy Marsh Oct 09 '20 at 13:55

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