What is the order of the set of distinct (up to similarity) nxn matrices over $\mathbb{R}$ with determinant equal to some non-zero scalar... say 6? (eg. countable, uncountable etc.)
The set of matrices over $\mathbb{R}$ is uncountable. So is the order of the set consisting of classes of matrices with the same determinant. In each of those classes we have further subsets, the equivalence classes formed by grouping similar matrices. Each equivalence class of similar matrices represents one linear transformation expressed in terms of all possible bases of $\mathbb{R}^n$, so the size of each equivalence class is also uncountable.
What I'm not certian about is the size of the set of "different" transformations that have the same determinant. Is it uncountable too?
What can I put it in a correspondence with to show this?
Apologies if this is poorly worded-- let me know if there is a better way to ask this question.