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I'm currently studying Cayley-Hamilton proof, but I think I'm missing something massive.

The theorem states that an endomorphism $\phi$ cancels its characteristic polynomial. I don't understand why the proof is not trivial.

Let $A$ be a matrix of $\phi$ in a basis $\mathcal{B}$. We know that the characteristic polynomial of $A$ is the same as $\phi$ no matter which basis of the vector space we take. Then, $$P_A(A)=\det(A-AI)=\det(A-A)=0$$

So the proof should be trivial, I don't understand where my error is.

Michael Hardy
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RFTexas
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    See this post. I find the second answer from the top particularly convincing. – Ben Grossmann Sep 07 '20 at 14:30
  • There is a formalism which allows the substitution to be made and constitute a proof. If I recall correctly, it is in Rings and Ideals (Carus Mathematical Monographs, No 8) (Carus Monograph) Textbook Binding – June 1, 1948 by Neal H McCoy (Author). – James S. Cook Sep 07 '20 at 14:35
  • It should be $\lambda I$ and $\lambda$ is a scalar, not a matrix. You can't make the substitution you've done of $A$ for $\lambda$ in this calculation as you have. – CyclotomicField Sep 07 '20 at 14:46

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