Exercise 7.2.18

Actually, this exercise could be a lemma for Exercise 7.2.17.

1.

It is nilpotent by Exercise 7.2.12 since $M$ is a upper triangular matrix with each diagonal entry equal to zero.

2.

It comes from some direct computation.

3.

Since $\mathit{MD}=\mathit{DM}$, we have

$$J^r={(M+D)}^r=\sum _{k=0}^r\left(\genfrac{}{}{0ex}{}{r}{k}\right)M^k{D}^{r-k}.$$

The second equality is due to the fact that $M^k=O$ for all $k\ge p$.