If $B \times B^T=I$, is $B^T \times B = I$?

Question

As described in the title, B is a rectangular (not square) matrix that has $m$ rows and $n$ columns. If $B \times B^T=I_{m \times m}$, is $B^T \times B = I_{n \times n}$ ?

Answer 1

If $B=(1 \ 0 ), BB^T=(1)=I_{1\times 1}$ and $B^T B=\begin{pmatrix}1 & 0 \cr 0 & 0\end{pmatrix}$

Answer 2

If $ B B^T=I_{m \times m}$, then $rank B=m$ and if $ B^T B=I_{n \times n}$, then $rank B^T=n$. But $rank B = rank B^T$, hence $n=m$.