RO-NMO 2003 Problem 114

Let be a $3\times 3$ real matrix $A.$ Prove the following statements.a) $f(A)\neq O_3,$ for any polynomials $f\in\mathbb{R} [X]$ whose roots are not real.b) $\exists n\in\mathbb{N} \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0$ Laurențiu Panaitopol