A=123123−1−2−3, then A is a nilpotent matrix of  index

# $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$, then A is a nilpotent matrix of  index

1. A

2

2. B

3

3. C

4

4. D

5

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### Solution:

${A}^{2}=\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& 3\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$\therefore$ A is a nilpotent matrix of index 2

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