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I'm stuck on the following:

Prove that if $A$ is an $n \times n$ real matrix then the the minimal polynomial of $A$ over the complex numbers has real coefficients.

I've covered characteristic polynomials, minimal polynomials, the Cayley-Hamilton theorem and Jordan normal form in the course so far.

Help would be much appreciated!

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If $A$, with real coefficients, satisfies the polynomial $p(X)=b_0+b_1X+\dots b_nX^n$ (with complex coefficients), then $A$ also satisfies $\bar{p}(X)=\overline{b_0}+\overline{b_1}X+\dots+\overline{b_n}X^n$, the polynomial where each coefficient is conjugate; therefore $A$ also satisfies $p(X)-\bar{p}(X)$.

What happens if $p$ is the minimal polynomial of $A$?

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Hint: Call the minimal polynomial $p_A$, so that $$p_A(A) = 0.$$ Take the conjugate of both sides of this equation.

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