Example 6

Suppose $n= 2$. Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be an element of $\mathbf{G}\mathbf{L}_2(D)$. The elements $a$ and $c$ of D are not both zero. We will give the determinant of A explicitly.

If $a$ is not zero, then we have

(24) $A = \begin{pmatrix} 1 & 0 \\ ca^{-1} & 1 \end{pmatrix}\begin{pmatrix} a & 0 \\ 0 & d-ca^{-1}b \end{pmatrix}\begin{pmatrix} 1 & a^{-1}b \\ 0 & 1 \end{pmatrix}$.

Consequently, $ad-aca^{-1}b\not= 0$ and

(25) det(A) $=\pi (ad-aca^{-1}b)$.

b) If $a$ is zero, then we have $c\not= 0$ and

(26) $A = \begin{pmatrix} 0 & b \\ c & d \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} c & d \\ 0 & b \end{pmatrix}$.

Consequently, by a) and Example 2, we have $cb\not= 0$ and

(27) det(A) $=\pi (-cb)$.