Given $\mathrm A$, we compute one Jordan decomposition, $\mathrm A = \mathrm P \mathrm J \mathrm P^{-1}$, where

$$\mathrm J = \begin{bmatrix} -1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\end{bmatrix}$$

Note that $\mathrm A^{30} = \mathrm P \mathrm J^{30} \mathrm P^{-1}$. Since $(-1)^{30} = 1$ and $\begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix}^{30} = \begin{bmatrix} 1 & \binom{30}{1}\\ 0 & 1\end{bmatrix} = \begin{bmatrix} 1 & 30\\ 0 & 1\end{bmatrix}$, we have

$$\mathrm J^{30} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 30\\ 0 & 0 & 1\end{bmatrix}$$

Thus,

$$\mathrm A^{30} = \mathrm P \mathrm J^{30} \mathrm P^{-1} = \cdots = \begin{bmatrix} 1 & 0 & 0\\ 15 & 1 & 0\\ 15 & 0 & 1\end{bmatrix}$$

Given $\mathrm A$, we compute one Jordan decomposition, $\mathrm A = \mathrm P \mathrm J \mathrm P^{-1}$, where

$$\mathrm J = \begin{bmatrix} -1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\end{bmatrix}$$

Note that $\mathrm A^{30} = \mathrm P \mathrm J^{30} \mathrm P^{-1}$. Since $(-1)^{30} = 1$ and $\begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix}^{30} = \begin{bmatrix} 1 & 30\\ 0 & 1\end{bmatrix}$, we have

$$\mathrm J^{30} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 30\\ 0 & 0 & 1\end{bmatrix}$$

Thus,

$$\mathrm A^{30} = \mathrm P \mathrm J^{30} \mathrm P^{-1} = \cdots = \begin{bmatrix} 1 & 0 & 0\\ 15 & 1 & 0\\ 15 & 0 & 1\end{bmatrix}$$