I am trying to prove that the geometric multiplicity of an eigenvector is bounded by the algebraic multiplicity. One particular proof of this theorem that I like is contained in the answer by Mariano Suárez-Alvarez on this Mathematics Stackexchange post. But there is one aspect of this proof that I do not understand, and I would like it if somebody would please explain it to me.
In the proof that I linked to, the author states that $t I-A$ has the same determinant as $C$. It is not obvious to me why this is the case. It certainly is not true in general that row-equivalent matrices $\mathbf A$ & $\mathbf B$ have the same determinant.
Meta: Rather than answering my current question here, perhaps it is better to edit the proof in the answer provided in the link.