Determinant of Kronecker Delta Elements

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Theorem

Let $\lambda$ and $\pi$ be permutations on $\left\{{1, 2, 3}\right\}$.

Let:

$\left({i, j, k}\right) = \left({\lambda \left({1}\right), \lambda \left({2}\right), \lambda \left({3}\right)}\right)$
$\left({r, s, t}\right) = \left({\pi \left({1}\right), \pi \left({2}\right), \pi \left({3}\right)}\right)$


Then:

$\begin{vmatrix} \delta_{ir} & \delta_{is} & \delta_{it} \\ \delta_{jr} & \delta_{js} & \delta_{jt} \\ \delta_{kr} & \delta_{ks} & \delta_{kt} \end{vmatrix} = \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right)$

where:

$\delta_{ir}$ denotes the Kronecker delta
$\begin{vmatrix} \cdot \end{vmatrix}$ denotes a determinant
$\operatorname{sgn} \left({i, j, k}\right)$ is the sign of the permutation $\left({i, j, k}\right)$ of the set $\left\{{1, 2, 3}\right\}$.


Proof


Sources