Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or $(3,1,2)$}\\ -1, & \text{if $(i,j,k)=(1,3,2)$, $(2,1,3)$ or $(3,2,1)$} \end{cases} $$
While I know $$|M|=\epsilon_{ijk}M_{1i}M_{2j}M_{3k}, \tag{*}\label{*}$$ I cannot answer the following question, which states:
From \eqref{*}, derive that $\epsilon_{pqr} |M|=\epsilon_{ijk}M_{pi}M_{qj}M_{rk}$
Could someone give a proof using suffix notation please!
Here are some breadcrumbs...
The key ingredient is: What happens to the determinant of a matrix when you swap two rows? What is the determinant of a matrix that has repeated rows?
Say we start with a matrix $M$ and want to compute $\epsilon_{ijk}M_{pi}M_{qj}M_{rk}$ (for $p,q,r$ distinct). Let $\tilde{M}$ be the matrix obtained by sliding row $p$ of $M$ up to row $1$, row $q$ up to row $2$, and row $r$ up to row $3$. Then $M_{pi}=\tilde{M}_{1i}$, etc. Use (*) to compute $|\tilde{M}|$. How are $|M|$ and $|\tilde{M}|$ related?
The case where $p,q,r$ are not distinct is a consequence of the fact that a matrix with repeated rows has zero determinant.