If a polynomial equation is satisfied by a matrix .Is the Polynomial it's characteristics equation????

Question

If a n order square matrix Satisfy any Arbitrary n degree Polynomial.. Can we say it is the character Polynomial Of the Matrix..???

If NO then what is necessary condition for the Matrix or Polynomial to be The statement true????

Answer 1

No. If $p(A)=0$ then the only thing you can say is that $\mu_A\mid p$, where $\mu_A$ is the minimal polynomial of $A$. In particular, the characteristic polynomial $\chi_A$ has the same property $\chi_A(A)=0$, so $\mu_A\mid\chi_A$.

There isn't much more you can say in general. However, for a lot of matrices (especially those that have all the eigenvalues different) you have $\mu_A=\chi_A$ so in that case you can replace $\mu_A\mid p$ with $\chi_A\mid p$.

In the latter case, you can still not conclude that $\chi_A=p$, though, without an additional information about $p$. For example, if, as in your example, $A$ is of order $n$ (so the degree of $\chi_A$ is $n$) and $p$ is also of degree $n$, then knowing $p\mid\chi_A$ lets you conclude that $p$ is "essentially" the same as $\chi_A$ (i.e. they are the same up to a constant multiplier).