For questions about the Cayley-Hamilton theorem, which states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.
If A is an $n×n$ matrix and $I_n$ is the $n×n$ identity matrix, the characteristic polynomial of A is defined as
$$ p(\lambda) = \det(\lambda I -A) $$
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation; that is, $p(A)=0$. This is deeper than it appears at first: note that this does not follow by putting $\lambda=A$, as that is an abuse of notation.
There are several proofs; one is to approximate $A$ by diagonal matrices and then invoke continuity. Another uses Nakayama's Lemma.
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