I'm trying to prove that the Levi-Civita symbol $\epsilon_{i_1 ... i_n}$ is a tensor density of weight $w=-1$. For this purpose, it has to be shown that the transformation law for the components of this pseudo-tensor under a change of basis, $\hat \epsilon_{j_1 ... j_n}$, is given by
$$\hat \epsilon_{j_1 ... j_n} =\big(\det(C)\big)^{-1} \epsilon_{i_1 ... i_n} C^{i_1}_{j_1}...C^{i_n}_{j_n} \tag{1}$$
With $C=(C^a_b)_{n \times n}$ being the change-of-basis matrix. I have seen in this related post that this is done using the expression of the determinant of a matrix through the Levi-Civita symbol:
$$ \tilde{\epsilon}_{\mu_1' \mu_2' \dots \mu_n'} |M| = \tilde{\epsilon}_{\mu_1 \mu_2 \dots \mu_n} M^{\mu_1}{}_{\mu_1'}M^{\mu_2}{}_{\mu_2'} \dots M^{\mu_n}{}_{\mu_n'} \tag{2.66}$$
However, I have not clear what is the relation of this expression and the expression of the determinant,
$$|M|=\epsilon_{i_1 ... i_n}M^{i_1}_{1}...M^{i_n}_{n} \tag{2}$$
How could equation (2) be modified in order to get (1) and complete the demonstration?