TITLE: Is the proof of this lemma really necessary?
QUESTION [18 upvotes]: To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:

Lemma: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$

The theorem (which says that if $f$ is an endomorphism of V, then $f$ is a solution to its characteristic polynomial), was then proven thus:
let $B(t) = \text{adj}[A-tI]$ and $P(t) = B(t)[A-tI]$, then $P(A)=0$ but also $P(t) = \delta I$ (where $\delta = \det(A-tI)$). Since $\delta = \chi_f(t)$, so $P(A) = 0 \Rightarrow \chi_f(A) = 0$.
My question is: since we interpret the $P(t)$ of the theorem as a polynomial with matrix coefficients, isn't the whole thing kind of obvious for the properties of a polynomial ring? (Assuming we all know how to switch between matrices and endomorphisms)

REPLY [4 votes]: I'd just like to make the following three observations as a minor complement to the other answers. 
OBSERVATION 1. What the Cayley-Hamilton Theorem says is $$\det
\begin{pmatrix}
a_{11}-A & a_{12}&\cdots&a_{1n}\\ 
a_{21}&a_{22}-A&\cdots&a_{2n}\\ 
\vdots&\vdots&\vdots&\vdots\\ 
a_{n1}&a_{n2}&\cdots&a_{nn}-A
\end{pmatrix}=0.$$
OBSERVATION 2. The proof of the Cayley-Hamilton Theorem I like best (among the ones I know) is on page 21 (proof of Proposition 2.4) of Introduction to Commutative Algebra by Atiyah and MacDonald. The argument can be phrased as follows.
Let $K$ be a commutative ring; let $n$ be a positive integer; let $A=(a_{ij})\in M_n(K)$ be an $n$ by $n$ matrix with entries in $K$; let $\chi$ be its characteristic polynomial; define $B=(b_{ij})\in M_n(K[A])$ by $b_{ij}:=\delta_{ij}\,A-a_{ij}$; observe $$\sum_i\ \ b_{ij}\ e_i=0,\quad\det B=\chi(A);$$ and write $(c_{ij})$ for the adjugate of $B$. Applying (a trivial case of) Fubini's Theorem to the double sum $\sum_{i,j}\ c_{jk}\ b_{ij}\ e_i$, we get $\chi(A)=0$. 
OBSERVATION 3. It's easy to define the ring $R[X]$ of polynomials in the indeterminate $X$ with coefficients in the noncommutative $R$. But when you think in terms of universal properties, you see that this construction is not very natural. So, it's better, I think, not to introduce it just to prove the Cayley-Hamilton Theorem.