Definition and condition of Nilpotent Matrix

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A matrix A is said to be Nilpotent Matrix if \[A^2=0\]. i.e square of matrix A is null or zero matrix .

For Example. Consider Matrix A=  \[ \begin{bmatrix} 1&2 &3 \\ 1 &2 &3 \\ -1 & -2 &-3 \\ \end{bmatrix}\] 

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