Strictly Triangular Matrices

If the entries on the diagonal of an upper (or lower) triangular matrix are all 0, the matrix is said to be strictly upper (or lower) triangular.
Example: The matrix

\[ \left| \begin{array}{ccc} 0 & 2 & 1 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{array} \right| \]

is strictly upper triangular and the matrix

\[ \left| \begin{array}{ccc} 0 & 0 & 0 \\ 2 & 0 & 0 \\ 3 & 0 & 0 \end{array} \right| \]

is strictly lower triangular.