Example: The matrix
\[ \left| \begin{array}{ccc} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right| \]
is atomic upper triangular and the matrix \[ \left| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 3 & 2 & 1 \end{array} \right| \]
is atomic lower triangular.The inverse of an atomic upper (or lower) triangular matrix takes the same form and can be instantly written down The inverses of the two matrices above are
\[ \left| \begin{array}{ccc} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right| \]
and \[ \left| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & -2 & 1 \end{array} \right| \]
is atomic lower triangular.