i have two claims i need to check and i'm not sure about.
claims are:
a)if a square matrix $A$ zeros(annihilates) the polynom$p(t)=t^2+5t+1$ then $A$ is invertible$
b)if a square matrix $A$ zeros(annihilates) the polynom$p(t)=t^{102}+t^2+t$ then $A$ is invertible
what i think:
a)not true, because for the matrix to be invertible, The equation $Ax=0$ has to have only the trivial solution $x=0$.
b)true, because in this case only x=0 zeros the matrix.
and a small question if i may, i saw a question that happens to be very simple and i'm wondering if it is that simple or a trick question(claim) i don't understand: $A=diag{0,0,0,0,0,0}$ then A zeros(annihilates) the polynom $t^6-t$. seems to be obviously true according to Cayley–Hamilton theorem. is it that simple or am i missing something?
thank you very much for your help.
Actually:
a) It's true, because $A^5+5A+\operatorname{Id}=0$ and therefore $A(-A^4-5\operatorname{Id})=\operatorname{Id}$.
b) Not true. The null matrix annihilates the polynomial.