A paradox is an argument that starts with apparently acceptable assumptions and leads by apparently valid deductions to an apparent contradiction. Since logic admits no contradictions, either the apparently acceptable assumptions are not acceptable, or the apparently valid
deductions are not valid, or the apparent contradiction is not a contradiction. A paradox moves us to reexamine the argument until we find out what is wrong.
Determined number r, that is, the largest order among the non-zero minor determinants of A, is called the rank of A.
Accordingly, there exists in a matrix of rank r at least one non-zero minor determinant of order r, while all minor determinants of order higher than r, if present, vanish.
Obviously, the rank of a matrix A does not change if you alter arbitrarily the sequence of its rows and columns, for the set of the minor determinants remains the same apart from changes in sign.
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