Minimal polynomial of an inverse given the min polynomial of the matrix

Question

Question: Given that $A$ is an invertible matrix and $m_A(x)=a_0+a_1x+...+a_nx^n$ find $m_{A^{-1}}$.

Thought: If I put A in the polynomial then it's equal to 0, then I multiply the entire equation by (A^-1)^n , then I get some kind of polynomial: $a_n+a_{n-1}x+...+a_0x^n$ - I know that $A^{-1}$ makes it become zero, but how do I know if it's indeed the minimal?

Let $m_{A^{-1}}$ be the minimal polynomial of $A^{-1}$. You know that divides $a_n + a_{n-1}x +\dotsc + a_0 x^n$. Now interchange the roles. — Daniel Fischer, Nov 26 '13 at 12:26

score 2 · Answer 1 · edited Nov 14 '20 at 16:38

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An exercise for you that may help you here:

If in some field $\;\Bbb F\;$ we have that $\;\alpha\neq 0\;$ is a root of $\;a_0+a_1x+\ldots+a_{n-1}x^{n-1}+a_nx^n\in\Bbb F[x]\;$ , then $\;\alpha^{-1}\;$ is a root of $\;a_n+a_{n-1}x+\ldots+a_1x^{n-1}+a_0x^{n}\;$.

edited Nov 14 '20 at 16:38

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answered Nov 26 '13 at 12:26

DonAntonio

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Could you give a hint as to why one can switch the scalars? – Alex Apr 02 '21 at 08:55
@Alex Do it: if $$a_0+a_1\alpha+\ldots+a_n\alpha^n=0$$ then multiply both sides by $;\alpha^{-1};$ and check what happens... – DonAntonio Apr 02 '21 at 11:35

Minimal polynomial of an inverse given the min polynomial of the matrix

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