Determinant of Kronecker Delta Elements

Theorem

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

Let:

$\tuple {i, j, k} = \tuple {\map \lambda 1, \map \lambda 2, \map \lambda 3}$

$\tuple {r, s, t} = \tuple {\map \pi 1, \map \pi 2, \map \pi 3}$

Then:

$\begin {vmatrix}

 \delta_{i r} & \delta_{i s} & \delta_{i t} \\
 \delta_{j r} & \delta_{j s} & \delta_{j t} \\
 \delta_{k r} & \delta_{k s} & \delta_{k t}

\end {vmatrix} = \map \sgn {i, j, k} \map \sgn {r, s, t}$

where:

$\delta_{ir}$ denotes the Kronecker delta

$\begin{vmatrix} \cdot \end{vmatrix}$ denotes a determinant

$\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$.

Proof

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Sources

• 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.2$: The summation convention