Example 7

Let $D^o$ be the opposite field of D. The mapping $A\mapsto^tA$ from $\mathbf{M}_n(D)$ to $\mathbf{M}_n(D^o)$ satisfies $^t($AB$) =^tB^tA$. Therefore, for every matrix A in $\mathbf{G}\mathbf{L}_n$(D), the transposed matrix $^tA$ is invertible in $\mathbf{M}_n(D^o)$, but it is not necessarily invertible in $\mathbf{M}_n(D)$. It follows from Example 3 that if A belongs to $\mathbf{G}\mathbf{L}_n$(D), then the elements A of $\mathbf{G}\mathbf{L}_n(D)$ and $^tA$ of $\mathbf{G}\mathbf{L}_n(D^o)$ have the same determinant. On the other hand, even if the matrix $^tA$ belongs to $\mathbf{G}\mathbf{L}_n$(D), its determinant in GL$_n(D)$ is not necessarily equal to that of A (cf. Example 6).