# Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:

“The rank of a matrix A is the number
of non-zero rows in the reduced
row-echelon form of A”.

The lecturer then explained that if the matrix $$AA$$ has size $$m×nm \times n$$, then $$rank(A)≤mrank(A) \leq m$$ and $$rank(A)≤nrank(A) \leq n$$.

The way I had been taught about rank was that it was the smallest of

• the number of rows bringing new information
• the number of columns bringing new information.

I don’t see how that would change if we transposed the matrix, so I said in the lecture:

“then the rank of a matrix is the same of its transpose, right?”

And the lecturer said:

“oh, not so fast! Hang on, I have to think about it”.

As the class has about 100 students and the lecturer was just substituting for the “normal” lecturer, he was probably a bit nervous, so he just went on with the lecture.

I have tested “my theory” with one matrix and it works, but even if I tried with 100 matrices and it worked, I wouldn’t have proven that it always works because there might be a case where it doesn’t.

So my question is first whether I am right, that is, whether the rank of a matrix is the same as the rank of its transpose, and second, if that is true, how can I prove it?

Thanks 🙂