Fix a nonnegative integer d, a field F, and a vector space V over F with dimension d + 1. Let T denote an invertible upper triangular matrix in Mat d+1 (F). Using T we construct three flags on V . We find a necessary and sufficient condition on T for these three flags to be totally opposite. In this case, we use these three totally opposite flags to construct a Billiard Array B on V . It is known that B is determined up to isomorphism by a certain triangular array of scalar parameters called the B-values. We compute these B-values in terms of the entries of T . We describe the set of isomorphism classes of Billiard Arrays in terms of upper triangular matrices.
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