Is it possible for a matrix to have no leading ones? So that its reduced row echelon form has no leading ones. Is this possible?
Would the empty set be the only possible solution to this problem?
Is it possible for a matrix to have no leading ones?
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$\begingroup$
matrices
matrix-equations
matrix-calculus
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2Zero matrix is a possibility. – 2017-01-18
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0This is definitely possible, and it does not have to be the zero matrix. Consider [[0,1],[0,0]] (I apologize for the formatting). – 2017-01-18
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2@user3798897 Your example doesn't work because the $1$ in the first row of your example is a leading one. – 2017-01-18
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0Note that the rank of a matrix equals the number of leading ones in its (reduced) row echelon form. – 2017-01-19
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0Thank you everyone for your help! I get it now. :) – 2017-01-21
3 Answers
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The zero matrix, of any size, is in reduced row echelon form with no leading ones. As soon as there is a nonzero entry, there will be at least a leading one.
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0Thank you very much for your help! – 2017-01-21
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If the reduced row echelon form has no leading ones, it is the zero matrix, and therefore any matrix which reduces to that is already the zero matrix.
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Not quite the empty set. The obvious answer is $A=0$.