I'm doing question $1.5$ in Matrix Groups for Undergrads and I came across this question:
where equaton $1.5$ is given by
Here is the definition of $R_A$.
I can see why the map $R_A$ is not invertible, because I can choose my 'right' vector, $X$ as $(-a,a)$ for any $a \in \mathbb{H}$ and I would get the zero vector in return (based on def $1.10$).
However, isn't the determinant based on equation $1.5$ going to give me $ij-ij=0$. I don't see how I can show the determinant is not equals to zero.