question about representation of the minimum polynomial Let $T$ be a linear operator on $R^3$ which is represented in the standard ordered
basis by the matrix
$\begin{pmatrix}
6  &-3 &-2  \\
 4 &- 1& -2  \\
 10 &- 5& -3  \\
\end{pmatrix}$
Express the minimal polynomial $p$ for $T$ in the form $p = p_1p_2$, where $p_1$ and $p_2$
are monic and irreducible over the field of real numbers. 
The characteristic polynomial is: $(x-2)(x^2+1)$. In this case who will be the minimum polynomial? The characteristic polynomial itself? 
I can not decompose more than this because exercise demands that they $p_1$ and $p_2$ be irreducible in $R$.

Well actually $X-2$ and $X^2+1$ are irreducible over $\mathbb R$.
As for the minimal polynomial, you know that it divides $(X-2)(X^2+1)$. Notice that $X-2$ and $X^2+1$ do not annihilate your matrix. Hence the minimial polynomial is the characteristic polynomial itself!
