Q:

Factorization of a linear combination of matrices

I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper)
http://www.cs.berkeley.edu/~wkahan/MathH110/DownDets.pdf
I have attached a picture of the point I don't understand:

I have higlighted the area I don't understand with a red rectangle. Could someone clarify why can we do the factorization and the last remark (about the injectivity) made by the author?
Thank you for any help :) 

A:

If we can factor a polynomial $p(z)$, we can also factor when $z$ is replaced by a matrix argument $T$.  Proving this fact is a simple but slightly messy induction argument (we induct on the degree of the polynomial).
The reason that at least one $T-r_jI$ is not injective is that the product of injective linear operators is always injective.  Thus if every $T-r_jI$ were injective, $c(T-r_1I)\cdots(T-r_mI)$ would also be injective - but this contradicts the construction of  $c(T-r_1I)\cdots(T-r_mI)$, since we have $c(T-r_1I)\cdots(T-r_mI)v=0$ for $v\neq 0$.

