\begin{aligned} & A^2=A A=\left(\begin{array}{lll} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array}\right)\left(\begin{array}{ccc} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array}\right)=\left(\begin{array}{ccc} \lambda^2 & 2 \lambda & 1 \\ 0 & \lambda^2 & 2 \lambda \\ 0 & 0 & \lambda^2 \end{array}\right) \\ & A^3=A^2 A=\left(\begin{array}{ccc} \lambda^2 & 2 \lambda & 1 \\ 0 & \lambda^2 & 2 \lambda \\ 0 & 0 & \lambda^2 \end{array}\right)\left(\begin{array}{ccc} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array}\right)=\left(\begin{array}{ccc} \lambda^3 & 3 \lambda^2 & 3 \lambda \\ 0 & \lambda^3 & 3 \lambda^2 \\ 0 & 0 & \lambda^3 \end{array}\right) \\ & A^4=A^3 A=\left(\begin{array}{ccc} \lambda^3 & 3 \lambda^2 & 3 \lambda \\ 0 & \lambda^3 & 3 \lambda^2 \\ 0 & 0 & \lambda^3 \end{array}\right)\left(\begin{array}{ccc} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array}\right)=\left(\begin{array}{ccc} \lambda^4 & 4 \lambda^3 & 6 \lambda^2 \\ 0 & \lambda^4 & 4 \lambda^3 \\ 0 & 0 & \lambda^4 \end{array}\right) \\ & \end{aligned}

$$\lambda, \lambda^2, \lambda^3, \lambda^4, 1,2 \lambda, 3 \lambda^2, 4 \lambda^3, 0,1,3 \lambda, 6 \lambda^2 .$$

$$\begin{gathered} \lambda, \lambda^2, \lambda^3, \lambda^4, \cdots, \lambda^n, 1,2 \lambda, 3 \lambda^2, 4 \lambda^3, \cdots, n \lambda^{n-1} \\ 0,1,3 \lambda, 6 \lambda^2, \cdots, \frac{n(n-1)}{2} \lambda^{n-2} \end{gathered}$$

$$A^k=\left(\begin{array}{ccc} \lambda^k & k \lambda^{k-1} & \frac{k(k-1)}{2} \lambda^{k-2} \\ 0 & \lambda^k & k \lambda^{k-1} \\ 0 & 0 & 0 \end{array}\right),$$

$$A^n=\left(\begin{array}{ccc} \lambda^n & n \lambda^{n-1} & \frac{1}{2} n(n-1) \lambda^{n-2} \\ 0 & \lambda^n & n \lambda^{n-1} \\ 0 & 0 & \lambda^n \end{array}\right) .$$

\begin{aligned} A^{n+1}=A^n A & =\left(\begin{array}{ccc} \lambda^n & n \lambda^{n-1} & \frac{n(n-1)}{2} \lambda^{n-2} \\ 0 & \lambda^n & n \lambda^{n-1} \\ 0 & 0 & \lambda^n \end{array}\right)\left(\begin{array}{lll} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array}\right) \\ & =\left(\begin{array}{ccc} \lambda^{n+1} & (n+1) \lambda^n & \frac{(n+1) n}{2} \lambda^{n-1} \\ 0 & \lambda^{n+1} & (n+1) \lambda^n \\ 0 & 0 & \lambda^{n+1} \end{array}\right) \end{aligned}

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