Solution
= Solution
Write $a=(a_1,a_2,a_3,a_4)^T$. The <reflection in a hyperplane> orthogonal to $a$ sends
$$
x\longmapsto x-2\frac{a^Tx}{a^Ta}a,
$$
so
$$
\boxed{M=I_4-\frac{2aa^T}{a^Ta}}.
$$
It acts as $-1$ on $\operatorname{span}\{a\}$ and as $1$ on the three-dimensional hyperplane $a^\perp$. Therefore
$$
\boxed{\det M=-1}.
$$