Minimal polynomial of upper triangular block matrix

Question

Let $T$ be a upper triangular block matrix

\begin{bmatrix} A & B \\ 0 & C \\ \end{bmatrix}

I want to show that the minimal polynomials of $A$ and $C$ divide the minimal polynomial of $T$. I know that it is true for the block diagonal matrix, but I don't know how to deal with $B$.

Thank you.

Answer 1

If $m$ denotes the minimal polynomial of your matrix and $p,q$ are the minimal polynomials of $A$ and $C$ respectively, then $$ m\left(\begin{bmatrix} A&B\\ 0&C\end{bmatrix}\right)=\begin{bmatrix} m(A)&*\\0&m(C)\end{bmatrix}, $$ so $m(A)=0$ and $m(C)=0$. This implies that $p|m$ and $q|m$.