Determinant of a 4 by 4 matrix

Question

Suppose $(a,b,c,d)\in \mathbb R^4 $ is nonzero. $ M = \left( {\begin{array}{cc} a & -b & -c &-d \\ b & a & -d & c \\ c &d &a &-b \\ d&-c&b&a \end{array} } \right) $.

Is det$(M)$ non-zero?

Answer 1

I checked, the way it is typed now agrees with $a (id) + b i + c j + d k$ in https://en.wikipedia.org/wiki/Quaternion#Matrix_representations so the determinant is $$ a^2 + b^2 + c^2 + d^2 $$ and is nonzero if any of the (real) entries is nonzero

The description as SU2, meaning $$ \left( \begin{array}{rr} \alpha & \beta \\ - \bar{\beta} & \bar{\alpha} \end{array} \right) $$ fits with the other (out of 48 possible) set of four matrices on wikipedia,