Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 12B / iv / Solution

Both sides are alternating in $i,j,k$ and in $l,m,n$. If either triple repeats an index, both sides vanish. Otherwise each triple is a permutation of $(1,2,3)$; the determinant of the corresponding permutation matrix is exactly the product of the two permutation signs. HenceThere are $3!=6$ nonzero components of ${ϵ}_{ijk}$, each equal to $\pm 1$, so