## Nilpotent Matrix and Eigenvalues of the Matrix

## Problem 11

An $n\times n$ matrix $A$ is called **nilpotent** if $A^k=O$, where $O$ is the $n\times n$ zero matrix.

Prove the followings.

**(a)** The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.

**(b)** The matrix $A$ is nilpotent if and only if $A^n=O$.

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