Question

why row rank of a matrix is equal to column rank of the matrix ?

Answer 1

if k  row of a matrix with n row is linearly independent then we say the rank of a matrix is k.

we know that if we take the transpose of a matrix the rank does nt changes so by taking the transpose row rank remains k =>so column rank is also k.(as now row and column is interchanged in the new matrix).

in simple word,by interchanging the row and column the linear dependency is not going to change. 

Comments

row ranks means number of linearly independent row of the matrix and column rank implies number of independent column of the matrix. For any matrix row rank = column rank = rank of the matrix.

My question was why this property  (matrix row rank = column rank ) holds for any matrix.

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https://math.stackexchange.com/questions/332908/looking-for-an-intuitive-explanation-why-the-row-rank-is-equal-to-the-column-ran/332945#332945?newreg=291dd42bbf7d445dbd3ef4006f6805d7

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Thanks for the link. But still I did not get why performing row operations won't change the column rank of the matrix. At least It'll change the column space of the matrix.