How to actually prove that the chracterstic polynomial is monic and of degree n

Question

So in my courses in linear algebra we consistently use that the characteristic polynomial is of degree n. ( that is assuming $f:V \to V$ with the dimension of $V$ being $n$) However, I have not actually see the proof of it. Does it just follow from simple definitions or is there something else one needs to notice?

Answer 1

The characteristic polynomial of a matrix $A$ is defined as $$p(t)=\det(A-tI)=\sum_{\sigma \in S_{n}}\epsilon(\sigma)\prod_{i=1}^{n}(a_{i \sigma(i)}-t\delta_{i \sigma(i)})$$ and the term in the sum with $\sigma=id$ has degree $n$.