find minimal polynomial of matrix product $AB$ where $AB=BA$ Let $x^2-1$ and $x^2+1$ be minimal polynomials of $A,B\in M_n(\mathbb{R})$, respectively.
If $AB=BA$, find the minimal polynomial of $AB$.

Notice that $A^2 = I$ and $B^2 = -I$. Therefore $(AB)^2 = A^2 B^2 = -I$, so the minimal polynom of $AB$ divides $x^2+1$. Because $x^2+1$ is monic and irreducible over $\mathbb{R}$ this already is the minimal polynomial.
