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$\begin{array}{l}A=\left[\begin{array}{lll}a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a\end{array}\right], \text { where } a \in \mathbb{F} \text {. Prove that } \\ A^{n}=\left[\begin{array}{ccc}a^{n} & n a^{n-1} & \frac{n(n-1)}{2} a^{n-2} \\ 0 & a^{n} & n a^{n-1} \\ 0 & 0 & a^{n}\end{array}\right] \quad \text { for all integers } n \geq 1 .\end{array}$

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