I am trying to solve some sample test questions and am looking for shortcut to a problem.
Question: True or False "Every invertible matrix $A \in \mathbb{Q}^{n \times n}$ is similar to a diagonal matrix over $\mathbb{C}$"
I think the statement is true for the following reason:
Theorem: Let $D$ be a principal ideal domain and $A \in D^{n\times n}$ then $A$ is equivalent to a matrix which has the diagonal form $diag\{d_a,d_2, \ldots, d_r, 0, \ldots, 0\}$ where the $d_i \neq 0$ and $d_i | d_j$ if $i \leq j$.
The proof for this fact has a lot of steps and requires one to induct using a suitably defined notion of "length" on a non-zero element in $D$. I was wondering if there is a simple trick to this problem by using invertibility of $A$.