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Unitriangular Matrices

If the entries on the diagonal of an upper or lower triangular matrix are all 1, the matrix is said to be upper (or lower) unitriangular.
Example: The matrix  
\[ \left| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array} \right| \]
  is upper unitriangular and the matrix  
\[ \left| \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 0 & 1 \end{array} \right| \]
  is lower unitriangular.

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