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Let A, B be square matrices of the same size. Then it can be proved that $det(\lambda I-AB)=det(\lambda I-BA)$, if we assumes that B is invertible.

I'm lost from here, since by definition, determinant is a number, then the identity $det(\lambda I-AB)=det(\lambda I-BA)$ shows two numbers are equal, but how can I conclude from it the polynomials are the same?

Thanks!

I found this solution https://math.stackexchange.com/a/3110428/533661, but I don't understand how specialize the entries of $A$ and $B$ to concrete values in the field of interest happens.

  • In this context the determinant is a polynomial in $\lambda$. – Dave May 11 '20 at 00:41
  • "Specialise entries to field" just means that $A$ and $B$ can have any values in the field you work with (e.g. $\mathbb{R}$, $\mathbb{C}$, etc.) – mi.f.zh May 11 '20 at 04:33

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