Unitriangular matrix power formula
= Unitriangular matrix power formula
{title2=$(I+A)^j=I+jA+\binom j2A^2\quad(A^3=0)$}
For a three-dimensional strictly upper triangular <matrix> $A$, the <binomial theorem> stops after $A^2$. Over a <finite field> of odd prime characteristic $p$, both coefficients $p$ and $\binom p2$ vanish, so every nonidentity <matrix> in $UT_3(\mathbb F_p)$ has order $p$. In characteristic two, a <matrix> with nonzero $A^2$ instead has order four.