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Which of the following conditions on an $n\times n$ matrix $A$ ensures that $A$ will be nilpotent?

A

All diagonal entries of $A$ are equal to $0$

B

$A$ is lower triangular

C

$A$ is idempotent

D

$A+I_n$ is upper triangular with all diagonal entries equal to $1$

E

$A$ is non-invertible

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