TITLE: Invertible matrices over noncommutative rings
QUESTION [50 upvotes]: 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.

REPLY [52 votes]: See: R.N. Gupta, Anjana Khurana, Dinesh Khurana, and T.Y. Lam, 
Rings over which the transpose of every invertible matrix is invertible; J. Algebra 322 (2009), no. 5, 1627–1636 (MR).
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.  …