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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? (a11a2,n1a1na21a2,n2a2,n10an1,1an1,200an100)
  • Solution

    The sign of the permutation corresponding to the antidiagonal depends on n. It is a permutation (n,n1,,1). It has n(n1)2 inversions, and its sign is (1)n(n1)2.

    |a11a2,n1a1na21a2,n2a2,n10an1,1an1,200an100|=(1)n(n1)2ni=1ai,n+1i

    The sign changes in two steps, that is +,,,+,+,,,.

Difficulty level: Moderate task
Reasoning task
Proving or derivation task
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