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$A$ is a 3x3 matrix. $A$ is called positive, denoted $A>0$, if every entry of $A$ is positive. Similarly one can define $A>=0$. Let

$$f(x)=x^6+x^4-x^3+x^2+x$$

Then

$$f(A)=A^6+A^4-A^3+A^2+A$$

Could you prove that $f(A)>=0$ whenever $A>0$?

I have tried many matrices on CAS, it is true. But I don't know how to prove it.

Thanks.

$$f(x)=x^6+x^4-x^3+x^2+x$$

Then

$$f(A)=A^6+A^4-A^3+A^2+A$$

Could you prove that $f(A)>=0$ whenever $A>0$?

I have tried many matrices on CAS, it is true. But I don't know how to prove it.

Thanks.

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