Let $A\in M\_m(R)$ be an invertible square matrix over a **noncommutative** ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?


The question popped up while working on a paper. We need to impose that the *transpose* of certain matrix of endomorphisms is invertible, and we wondered if that was the same as asking if the matrix is invertible.



Invertible matrices over noncommutative rings
See: R.N. Gupta, Anjana Khurana, Dinesh Khurana, and T.Y. Lam, 
[Rings over which the transpose of every invertible matrix is invertible](http://www.sciencedirect.com/science/article/pii/S0021869309003482); J. Algebra 322 (2009), no. 5, 1627–1636 ([MR](http://www.ams.org/mathscinet-getitem?mr=2543626)).


Abstract: We prove that the transpose of every invertible square matrix over a ring $R$ is invertible if and only if $R/\text{rad}(R)$ is commutative. …


