Lemma 10.16.1. Let $R$ be a ring. Let $A = (a_{ij})$ be an $n \times n$ matrix with coefficients in $R$. Let $P(x) \in R[x]$ be the characteristic polynomial of $A$ (defined as $\det (x\text{id}_{n \times n} - A)$). Then $P(A) = 0$ in $\text{Mat}(n \times n, R)$.

Proof. We reduce the question to the well-known Cayley-Hamilton theorem from linear algebra in several steps:

1. If $\phi :S \rightarrow R$ is a ring morphism and $b_{ij}$ are inverse images of the $a_{ij}$ under this map, then it suffices to show the statement for $S$ and $(b_{ij})$ since $\phi$ is a ring morphism.

2. If $\psi :R \hookrightarrow S$ is an injective ring morphism, it clearly suffices to show the result for $S$ and the $a_{ij}$ considered as elements of $S$.

3. Thus we may first reduce to the case $R = \mathbf{Z}[X_{ij}]$, $a_{ij} = X_{ij}$ of a polynomial ring and then further to the case $R = \mathbf{Q}(X_{ij})$ where we may finally apply Cayley-Hamilton.

$\square$

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