Let be $K$ a field and $m, n \in \mathbb N_0$.
For a matrix $A\in K^{n \times n}$ it holds that: $A \in GL_n(K)$ if $A^{tr} \equiv E_n$. Where $E_n$ is the identity matrix.
Question: Is that statement true?
I think that it is true because a matrix is invertible if its transpose is invertible, too.
And by definition of equivalence of matrices:
Let $P \in GL_n(K) \quad Q\in GL_m(K)$
$A \equiv B$, if
$Q^{-1}A^{tr}P = B$
Where $Q$ is an identity matrix and $P$ inverse to $A^{tr}$.
Is that correct? Would that hold a complete proof?