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Vuldoraq

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**Proving the Cayley-Hamilton Theorem**

Hey all,

I'm revising for my linear algebra exam, which is next week, and I got up to the Cayley-Hamilton theorem, but I am stuck on the final leap in the proof. Here is what I understand so far,

Theorem:

Every matrix is a zero of it's characteristic equation[tex][P_{A}(\lambda)=det(A-\lambda I)][/tex],

For [tex]\\\\\\\\\\\\\ A \in M_{n}(k)[/tex] [tex] P_{A}(A)=0[/tex]

Proof:

[tex]

Let\ P_{A}(\lambda)=det(A-\lambda I)=\lambda^{n}+a_{n-1}\lambda^{n-1}+\ldots+a_{1}\lambda+a_{0}[/tex]

and consider,

[tex]\phi(\lambda)=adj(A-\lambda I)=B_{n-1}\lambda^{n-1}+ldots+B_{1}\lambda+B_{0}[/tex]

where [tex] B_{i} \in M_{n}(k)[/tex]

Given that for any [tex]C \in M_{n}(k); \\ C*adj(C)=det(C)*I.[/tex]

So by letting [tex] C=A-\lambda I[/tex] we have,

[tex](A- \lambda I)*\phi(\lambda)=det(A-\lambda I)I=P_{A}(\lambda)I[/tex]

Expanding we have,

[tex](A- \lambda I)*(B_{n-1}\lambda^{n-1}+ldots+B_{1}\lambda+B_{0})=(\lambda^{n}+a_{n-1}\lambda^{n-1}+ldots+a_{1}\lambda+a_{0})*I[/tex]

The next step has me in tears, my book says compare coefficients and add but I can't see how you would compare these? Please can anyone help me to complete this?

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