Nilpotent Matrices

A nilpotent matrix is a matrix  
\[\mathbf{M}\]
  such that  
\[\mathbf{M}^n = \mathbf{0}\]
  for some n. A nilpotent matrix must be a square matrix, else could not find  
\[\mathbf{M}^n = \mathbf{0}\]
  and it must have zero determinant, sine  
\[det (\mathbf{M}^n) =(det(\mathbf{M}))^n =0\]

Example:  
\[ \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right) \]
  is nilpotent since  
\[ \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right) = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\]

but
\[ \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \]
  is not nilpotent since
\[ \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right)\]

A nilpotent matrix must have zero determinant, but it does not follow that a matrix with zero determinant is nilpotent. The matrix above is a counterexample.

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