The question in short is, why the rank of a matrix is equal to the rank of its transpose? Matrix is an array of numbers. Then it's amazing to me that the number of linear independent rows coincides with the number of linear independent columns. I tried to find some fundamental answer to this question, which does not resort to concepts like singular values or eigenvalues, so that it can be explained to those who do not know linear algebra. Is there an elementary way to explain this fact?