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$\bullet$ The sign switch is familiar from complex numbers:

The regular representation of $\mathbb{C}$ over $\mathbb{R}$ is the embedding of $\mathbb{R}$-algebras $\mathbb{C} \to M_2(\mathbb{R})$ defined by $a+ib \mapsto \begin{pmatrix} a & -b \\ b & a \end{pmatrix}$. The inverse of $a+ib$ is the conjugate $a-ib$ devided divided by the norm $a^2+b^2$, thus the inverse of $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ is the adjugate $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ divided by the determinant $a^2+b^2$.

$\bullet$ Both the sign switch and the swap of the diagonal elements can be illustrated with quaternions:

The regular representations of $\mathbb{H}$ over $\mathbb{C}$ is the embedding $\mathbb{H} \to M_2(\mathbb{C})$ mapping $u+jv \mapsto \begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$. The inverse of $u + jv$ is the conjugate $\overline{u} - j \overline{v}$ divided by the norm $|u|^2+|v|^2$. Thus the inverse of $\begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$ is the adjugate \begin{pmatrix} \overline{u} & -v \\ \overline{v} & u \end{pmatrix} divided by the determinant $|u|^2+|v|^2$.

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$\bullet$ The sign switch is familiar from complex numbers:

The regular representation of $\mathbb{C}$ over $\mathbb{R}$ is the embedding of $\mathbb{R}$-algebras $\mathbb{C} \to M_2(\mathbb{R})$ defined by $a+ib \mapsto \begin{pmatrix} a & -b \\ b & a \end{pmatrix}$. The inverse of $a+ib$ is the conjugate $a-ib$ devided by the norm $a^2+b^2$, thus the inverse of $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ is the adjugate $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ devided divided by the determinant $a^2+b^2$.

$\bullet$ Both the sign switch and the swap of the diagonal elements can be illustrated with quaternions:

The regular representations of $\mathbb{H}$ over $\mathbb{C}$ is the embedding $\mathbb{H} \to M_2(\mathbb{C})$ mapping $u+jv \mapsto \begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$. The inverse of $u + jv$ is the conjugate $\overline{u} - j \overline{v}$ divided by the norm $|u|^2+|v|^2$. Thus the inverse of $\begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$ is the adjugate \begin{pmatrix} \overline{u} & -v \\ \overline{v} & u \end{pmatrix} divided by the determinant $|u|^2+|v|^2$.

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$\bullet$ The sign switch is familiar from complex numbers:

The regular representation of $\mathbb{C}$ over $\mathbb{R}$ is the embedding of $\mathbb{R}$-algebras $\mathbb{C} \to M_2(\mathbb{R})$ defined by $a+ib \mapsto \begin{pmatrix} a & -b \\ b & a \end{pmatrix}$. The inverse of $a+ib$ is the conjugate $a-ib$ devided by the norm $a^2+b^2$, thus the inverse of $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ is the adjugate $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ devided by the determinant $a^2+b^2$.

$\bullet$ Both the sign switch and the swap of the diagonal elements can be illustrated with quaternions:

The regular representations of $\mathbb{H}$ over $\mathbb{C}$ is the embedding $\mathbb{H} \to M_2(\mathbb{C})$ mapping $u+jv \mapsto \begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$. The inverse of $u + jv$ is the conjugate $\overline{u} - j \overline{v}$ divided by the norm $|u|^2+|v|^2$. Thus the inverse of $\begin{pmatrix} u & v \\ - \overline{v} & \overline{u} \end{pmatrix}$ is the adjugate \begin{pmatrix} \overline{u} & -v \\ \overline{v} & u \end{pmatrix} devided divided by the determinant $|u|^2+|v|^2$.