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Are there any cases when the row-rank and column rank of a matrix can not be equal? Or are they equal under all circumstances?

I understand the proof that they are equal but was wondering if there are any special cases when they are not equal.

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  • $\begingroup$ @DietrichBurde I understand they are equal and also understand the basic idea/proof, but my question was if there exist a special case/matrix when they are not equal, thanks. $\endgroup$ Commented Nov 4, 2017 at 9:43
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    $\begingroup$ Check out the proof again What are the assumptions ? if one of the assumptions are not satisfied, the result will not hold in general. $\endgroup$
    – Our
    Commented Nov 4, 2017 at 9:48

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They are always equal. That's a basic theorem in Linear Algebra. You will find a proof here, for instance.

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Yes, they are always equal.

Row rank is equal to the number of non-zero rows in the RREF while the column rank is equal to the number of pivot columns in the RREF.

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