The Cayley-Hamilton theory states that every matrix $A \in M_n(\mathbb{C})$ is a root of its own characteristic polynomial $p_A(x)$. My thought is this,
since by definition $p_A(x) = \det(xI-A)$, filling in $A$ into this polynomial we obtain
$$
p_A(A) = \det(AI-A) = \det(A-A) = \det(0) = 0.
$$
What is wrong with this reasoning?
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4Check this wikipedia link. – Vinícius Novelli Sep 16 '18 at 20:28
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Thanks! I wasn't aware of that one. I guess filling out matrices in polynomials has always kind of confused me. – Sigurd Sep 16 '18 at 20:32
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1$p_A(A)$ should turn out to be the null matrix $O$, whereas you obtain the scalar $0$. – StubbornAtom Sep 16 '18 at 20:39
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True, that seemed weird to me indeed. – Sigurd Sep 16 '18 at 20:41