Matrices representing symmetries

Question

Is there a direct way to find the matrices representing the symmetries for example of a tetrahedron with vertices $(1, 1 ,1)$ , $(-1, -1, 1)$, $(-1, 1 ,-1)$ , $(1, -1 ,-1)$

Answer 1

A symmetry of the tetrahedron is determined by where it takes any three of the vertices of the tetrahedron. Choose three vertices $v_1$, $v_2$ and $v_3$. Let $\sigma$ be a symmetry of the tetrahedron. Then the matrix that maps $v_i$ to $\sigma(v_i)$ is given by $$M_{\sigma}=\begin{pmatrix}\mid&\mid&\mid\\ \sigma(v_1)&\sigma(v_2)&\sigma(v_3)\\ \mid&\mid&\mid\end{pmatrix}\cdot\begin{pmatrix}\mid&\mid&\mid\\ v_1&v_2&v_3\\ \mid&\mid&\mid\end{pmatrix}^{-1}.$$ Here the $\sigma(v_i)$ and the $v_i$ form the columns of the matrices. As $\sigma$ and $M_{\sigma}$ agree on the $v_i$, they are the same. This requires calculating only one inverse, and a few matrix multiplications to get all matrices. For example, with $v_1:=(1,-1,-1)$, $v_2:=(-1,1,-1)$ and $v_3:=(-1,-1,1)$ we get $$\begin{pmatrix}\mid&\mid&\mid\\ v_1&v_2&v_3\\ \mid&\mid&\mid\end{pmatrix}^{-1}=\frac12\begin{pmatrix}0&-1&-1\\ -1&0&-1\\ -1&-1&0\end{pmatrix}.$$