Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 1 / b (Triangular powers)

If $A$ is a strictly upper triangular matrix, then $A_{ij}=0$ unless $j-i\ge 1$. We prove by mathematical induction thatThe assertion for $k=1$ is the definition. In the matrix multiplication formulaa nonzero summand would require $\ell -i\ge k$ and $j-\ell \ge 1$, hence $j-i\ge k+1$. This proves the induction step. Since $j-i\le n-1$ for all entries, $\boxed{A^n=0}$. In particular, every strictly upper triangular matrix is a nilpotent matrix.

Write $A=I_3+N$, whereThe binomial theorem for commuting matrices applies because $I_{3}N=N{I}_3$. All terms of degree at least three vanish, so for every positive integer $n$,The same expression directly gives $A$ when $n=1$, so no small exponent is excluded.