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

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

Solution:

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

$∴$ A is a nilpotent matrix of index 2

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