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Determinant of a triangular matrix
Task number: 4433
What is the formula for the determinant of triangular matrices of the following type?
(a11……a2,n−1a1na21…a2,n−2a2,n−10⋮⋮an−1,1an−1,20…0an10……0)
Solution
The sign of the permutation corresponding to the antidiagonal depends on n. It is a permutation (n,n−1,…,1). It has n(n−1)2 inversions, and its sign is (−1)n(n−1)2.
|a11……a2,n−1a1na21…a2,n−2a2,n−10⋮⋮an−1,1an−1,20…0an10……0|=(−1)n(n−1)2n∏i=1ai,n+1−iThe sign changes in two steps, that is +,−,−,+,+,−,−,⋯.