<s>[INST] For this matrix $A$, what is $A^n$? $$A = \begin{pmatrix}0 & a & b \\ 0& 0 & c \\ 0& 0 &0\end{pmatrix}$$
What is $A^n$ (for $n\geq 1)$?
 [/INST] This matrix is a nilpotent one, so we know that $A^3=0$. Now we calculate 
$$A^2=A\cdot A = \begin{pmatrix} 0 & a & b \\ 0 & 0& c \\0  & 0 & 0 \end{pmatrix}\cdot 
\begin{pmatrix} 0 & a & b \\ 0 & 0& c \\0  & 0 & 0 \end{pmatrix}=
\begin{pmatrix} 0 & 0 & ac \\ 0 & 0& 0 \\0  & 0 & 0 \end{pmatrix}$$
As $A^3=0$ $A^n=0$ for $n>2$
Instead of using that every strict upper triagle matrix is nilpotent we could calculate $A^3$
$$A^3 = A \cdot A^2= \begin{pmatrix} 0 & a & b \\ 0 & 0& c \\0  & 0 & 0 \end{pmatrix} \cdot
\begin{pmatrix} 0 & 0 & ac \\ 0 & 0& 0 \\0  & 0 & 0 \end{pmatrix}=
\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0& 0 \\0  & 0 & 0 \end{pmatrix} $$
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