## Atomic Triangular Matrices

If the entries on the diagonal of an upper (or lower) triangular matrix are all 1, and all other entries are zero except for a single row or column above (or below) the main diagonal the matrix is said to be atomic upper (or lower) triangular.
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.