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A square matrix of integer coefficient is called unimodular, if its determinant is equal to \pm 1.

One can use LU-decomposition to generate random unimodular matrices: first generate 3 random matrices A,B,C, where A and C are square triangular unimodular matrices, one is upper-triangular, the other lower-triangular, and B is a permutation matrix (that is, every row and every column is composed of one 1 and the other are 0's). Then the product ABC gives a random unimodular matrix.