Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 1 / b / i / Solution

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.