Example 3

Suppose $n\geqslant 1$. For every invertible diagonal matrix of the form Δ = diag($d_1, . . . , d_n$), we have Δ$B_{ij}(\lambda$)Δ$^{-1}=B_{ij}(d_i\lambda d^{-1}_j)$. Let A be an element of $\mathbf{G}\mathbf{L}_n(D)$. By Corollary 1 of II, §10, No. 13, p. 362 and the previous formula, there exist matrices P and Δ in $\mathbf{G}\mathbf{L}_n(D)$ such that A = PΔ, that P is the product of matrices of the form $B_{ij}(\lambda )$, and that Δ is a diagonal matrix of the form diag(1$, . . . ,1, d$). We have det(P) = 1 by Example 1, and therefore det(A) = det(Δ) $=\pi (d)$ by Proposition 3.