Upper triangular matrix and nilpotent

Question

How to show that an upper triangular matrix is nilpotent if and only if all its diagonal elements are equal to zero by using induction?

Answer 1

Let $A$ be a upper triangular matrix of order $n$ and let us denote $a_{ij}^{(k)}$ the $(i, j)$ element of $A^k$.

• If $a_{ii}^{(1)} \neq 0$ for some $1 \leqslant i \leqslant n$ then you can prove by induction that $a_{ii}^{(k)} \neq 0$ for every $k \geqslant 1$.
• If all diagonal elements of $A$ are equal to zero, then you can prove by induction that for every $k \geqslant 2$, all elements within the $k - 1$-diagonal of the matrix $A^k$ (i.e. elements $a_{ij}^{(k)}$ such that $j = i + k$) are equal to zero.