Is $\operatorname{rank}(A) = \operatorname{rank}( A^T )$ always true?

Question

The answer key for my exam says that it's not always true, but I can't seem to come up with an explanation for this or find a matrix that makes this equation untrue. Thanks

Answer 1

Rank refers to the row rank or column rank of a matrix. Row rank is the number of linearly independent rows in the matrix, while column rank refers to the number of linearly independent columns.
But, row rank of a matrix is equal to the column rank. So, transposing the matrix has no effect on the rank of the matrix.

Answer 2

Its simple to understand, row operation leave the both the row and column rank unchanged. Same with column operations. By using row and column operations together one can reduce the matrix to diagonal form $$\begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\end{pmatrix}$$ For such a matrix the row and column rank are obviously the same.