Collection of Maths Problems

Determinant of a triangular matrix

Task number: 4433

• Solution

  The sign of the permutation corresponding to the antidiagonal depends on $n$. It is a permutation $(n,n-1,\dots,1)$. It has $\frac{n(n-1)}2$ inversions, and its sign is $(-1)^{\frac{n(n-1)}2}$.

  $$\begin{vmatrix} a_{11} & \dots & \dots & a_{2,n-1} & a_{1n} \\ a_{21} & \dots & a_{2,n-2} & a_{2,n-1} & 0 \\ \vdots & & & & \vdots \\ a_{n-1{,}1} & a_{n-1{,}2} & 0 & \dots& 0 \\ a_{n1} & 0 & \dots & \dots & 0 \end{vmatrix} = (-1)^{\frac{n(n-1)}2} \prod\limits_{i=1}^n a_{i,n+1-i}$$

  The sign changes in two steps, that is $+,-,-,+,+,-,-,\cdots$.