1. ## Cofactor Expansion

Let A be an n x n matrix.
a) Show that f(t) = det(tIn - A) is a polynomial in t of degree n.
b) What is the coefficient of t^n in f(t)?
c) What is the constant term in f(t)?

I don't understand how I would show it in a simpler way, than the way
that I am currently doing. I just need feedback about the way I'm
solving the problem!

I know that tIn would be an identity matrix multiplied by t. If I
subtract A from tIn than I would get a matrix like this:

|t - a11 ... -a1n|
|-a21 ...... -a2n|
|-an1 ... t - ann|

For the first part, I understand how the determinant is a polynomial
in the form of t^n.
The first part of the determinant that would be calculated would be
the diagonal, (a11)(a22)(a33)..(ann). That would give me the equation
(t-a11)(t-a22)...(t-a33).
The resulting equation would end in
t^n +- .... +- a11a22a33..ann. My first question is, is there an
easier way to prove this, or is this good enough.

For part b, I'm not sure if I was supposed to come up with a different
coefficient, besides 1. I did some examples with n = 3,4,5 and I
always got that the coefficient was 1.

For part c I got that the constant from part a (+- a11a22..ann) would
be added to the opposite diagonal (+-a1n a2n-1...an1).

I'm not sure if I did that right! Thanks so much for your help!

2. By proving this by induction on $n$ when you expand along on of the cofactors.