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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?

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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.
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