Let \(\displaystyle D=det(a_{ij})\)

\(\displaystyle \epsilon_{ijk}\epsilon_{pqr}D=

\begin{vmatrix}

a_{ip} & a_{iq} & a_{ir} \\

a_{jp} & a_{jq} & a_{jr} \\

a_{kp} & a_{kq} & a_{kr}

\end{vmatrix}

\)

The book demonstrates this way:

*"If (at least) two of i, j, k or two of p, q, r are equal, then both sides of (1.7.21) are 0. If i, j, k and p, q, r are both cyclic or acyclic, then each side of (1.7.21) is equal to D. If i, j, k are cyclic but p, q, r are acyclic or vice versa, then each side of (1.7.21) is equal to -D. Thus, in all possible cases, result (1.7.21) is verified."*

But for me it doesn't make sense, this sum would give 0.