Is the minimal polynomial of a triangular block matrix the lcm of the blocks' minimal polynomials?

Question

I know that the minimal polynomial of a diagonal block matrix: $diag\{A_1, ..., A_n\}$ is $lcm(M_{A_1}(\lambda), ... M_{A_n}(\lambda))$, but is this also true for triangular block matrices?

Thanks!

Answer 1

The answer is no. As an example, consider the $4 \times 4$ matrix $$ A = \pmatrix{0 & I_2\\0 & 0}. $$ The minimal polynomial of each block is $p(\lambda) = \lambda$, but the minimal polynomial of $M$ is $p_A(\lambda) = \lambda^2$.