\begin{aligned} & A^2=A A=\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right)\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right)=\left(\begin{array}{cccc} 3 & 2 & 1 & 0 \\ 6 & 4 & 2 & 0 \\ 9 & 6 & 3 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \\ & A^3=A^2 A=\left(\begin{array}{cccc} 3 & 2 & 1 & 0 \\ 6 & 4 & 2 & 0 \\ 9 & 6 & 3 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right)\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right)=\left(\begin{array}{cccc} 0 & 0 & 0 & 10 \\ 0 & 0 & 0 & 20 \\ 0 & 0 & 0 & 30 \\ 30 & 20 & 10 & 0 \end{array}\right) \\ & A^4=A^3 A=\left(\begin{array}{cccc} 0 & 0 & 0 & 10 \\ 0 & 0 & 0 & 20 \\ 0 & 0 & 0 & 30 \\ 30 & 20 & 10 & 0 \end{array}\right)\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right)=\left(\begin{array}{cccc} 30 & 20 & 10 & 0 \\ 60 & 40 & 20 & 0 \\ 90 & 60 & 30 & 0 \\ 0 & 0 & 0 & 100 \end{array}\right) \\ & \end{aligned}

\begin{aligned} A^5=A^4 A & =\left(\begin{array}{cccc} 30 & 20 & 10 & 0 \\ 60 & 40 & 20 & 0 \\ 90 & 60 & 30 & 0 \\ 0 & 0 & 0 & 100 \end{array}\right)\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right) \\ & =\left(\begin{array}{cccc} 0 & 0 & 0 & 100 \\ 0 & 0 & 0 & 200 \\ 0 & 0 & 0 & 300 \\ 300 & 200 & 100 & 0 \end{array}\right) \\ A^6=A^5 A & =\left(\begin{array}{cccc} 0 & 0 & 0 & 100 \\ 0 & 0 & 0 & 200 \\ 0 & 0 & 0 & 300 \\ 300 & 200 & 100 & 0 \end{array}\right)\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right) \\ & =\left(\begin{array}{cccc} 300 & 200 & 100 & 0 \\ 600 & 400 & 200 & 0 \\ 900 & 600 & 300 & 0 \\ 0 & 0 & 0 & 1000 \end{array}\right) \end{aligned}

\begin{aligned} A^1 & =\left(\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 3 & 2 & 1 & 0 \end{array}\right), A^3=\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \times 10^1 \\ 0 & 0 & 0 & 2 \times 10^1 \\ 0 & 0 & 0 & 3 \times 10^1 \\ 3 \times 10^1 & 2 \times 10^1 & 1 \times 10^1 & 0 \end{array}\right) \\ A^5 & =\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \times 10^2 \\ 0 & 0 & 0 & 2 \times 10^2 \\ 0 & 0 & 0 & 3 \times 10^2 \\ 3 \times 10^2 & 2 \times 10^2 & 1 \times 10^2 & 0 \end{array}\right) \end{aligned}
$$, \cdots, A^{2 k+1}=\left|\begin{array}{cccc} 0 & 0 & 0 & 1 \times 10^k \\ 0 & 0 & 0 & 2 \times 10^k \\ 0 & 0 & 0 & 3 \times 10^k \\ 3 \times 10^k & 2 \times 10^k & 1 \times 10^k & 0 \end{array}\right|$$

\begin{aligned} & A^2=\left(\begin{array}{cccc} 3 & 2 & 1 & 0 \\ 6 & 4 & 2 & 0 \\ 9 & 6 & 3 & 0 \\ 0 & 0 & 0 & 10^1 \end{array}\right), A^4=\left(\begin{array}{cccc} 3 \times 10^1 & 2 \times 10^1 & 1 \times 10^1 & 0 \\ 6 \times 10^1 & 4 \times 10^1 & 2 \times 10^1 & 0 \\ 9 \times 10^1 & 6 \times 10^1 & 3 \times 10^1 & 0 \\ 0 & 0 & 0 & 10^2 \end{array}\right) \\ & A^6=\left(\begin{array}{cccc} 3 \times 10^2 & 2 \times 10^2 & 1 \times 10^2 & 0 \\ 6 \times 10^2 & 4 \times 10^2 & 2 \times 10^2 & 0 \\ 9 \times 10^2 & 6 \times 10^2 & 3 \times 10^2 & 0 \\ 0 & 0 & 0 & 10^3 \end{array}\right) \\ &, \cdots, A^{2 k}=\left(\begin{array}{cccc} 3 \times 10^{k-1} & 2 \times 10^{k-1} & 1 \times 10^{k-1} & 0 \\ 6 \times 10^{k-1} & 4 \times 10^{k-1} & 2 \times 10^{k-1} & 0 \\ 9 \times 10^{k-1} & 6 \times 10^{k-1} & 3 \times 10^{k-1} & 0 \\ 0 & 0 & 0 & 10^k \end{array}\right) \end{aligned}

$$A^n=\left(\begin{array}{cccc} 0 & 0 & 0 & 10^{\frac{n-1}{2}} \\ 0 & 0 & 0 & 2 \cdot 10^{\frac{n-1}{2}} \\ 0 & 0 & 0 & 3 \cdot 10^{\frac{n-1}{2}} \\ 3 \cdot 10^{\frac{n-1}{2}} & 2 \cdot 10^{\frac{n-1}{2}} & 10^{\frac{n-1}{2}} & 0 \end{array}\right) .$$

$$A^n=\left(\begin{array}{cccc} 3 \cdot 10^{\frac{n}{2}-1} & 2 \cdot 10^{\frac{n}{2}-1} & 10^{\frac{n}{2}-1} & 0 \\ 6 \cdot 10^{\frac{n}{2}-1} & 4 \cdot 10^{\frac{n}{2}-1} & 2 \cdot 10^{\frac{n}{2}-1} & 0 \\ 9 \cdot 10^{\frac{n}{2}-1} & 6 \cdot 10^{\frac{n}{2}-1} & 3 \cdot 10^{\frac{n}{2}-1} & 0 \\ 0 & 0 & 0 & 10^{\frac{n}{2}} \end{array}\right) .$$