Minimal polynomial and characteristic polynomial of $T$

Question

1. Let be a vector space over the field and be a linear operator on . Then all eigen values of are zeros of the minimal polynomial of .
2. Minimal polynomial divides the characteristic polynomial.

Qus: If every zero of char. polynomial is a zero of minimal polynomial then how can minimal polynomial divide char. polynomial?

Note- A minimal polynomial $m(x)$ of $T$ is the monic polynomial with least degree s.t $m(T)=O$. Since $m(x)$ is of least degree, then how can it contain every zero of char. polynomial?

Answer 1

For $I_{3\times 3}$, characteristic polynomial is $p(x)=(x-1)^3$ while minimal polynomial is $m(x)=(x-1)$. Both polynomials have zeros at $x=1$ (eigenvalue of $I$), though the only possible difference is in their orders where it may be more for $p(x)$ than that of $m(x)$.