## 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.