
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
(ta11)(ta22)...(ta33).
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 a2n1...an1).
I'm not sure if I did that right! Thanks so much for your help!

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