Prove the characteristic equation is of degree n

Question

I'm attempting to prove the following theorem.

Let $A \in M_{n \times n}(F)$

The characteristic polynomial of $A$ is a polynomial of degree $n$ with leading coefficient $(-1)^n$

The theorem itself is very intuitive but I struggle handling all the indices when working with determinants and do not have much determinant theory down. Thanks for your help!

Answer 1

Think how you compute the characteristic polynomial. The highest order term will come from the product of the diagonal elements of $\mathbf{A}-\lambda \mathbf{I}$. The latter would be a product of the form $\prod_{i=1}^{n}(A_{ii}-\lambda)$ which is equal to $(-1)^{n}\lambda^{n} + \text{lower order terms}$.

Answer 2

The way is to work by induction on degree of successive polynomials you'd get by computing determinants.

Terry Tao gives a proof here. It's quite well-written and the same proof I had studied when I learnt Linear Algebra.