= Levi-Civita formulas for a three-dimensional determinant and inverse
{c}
{title2=$\det A=\frac16\epsilon_{ijk}\epsilon_{lmn}A_{il}A_{jm}A_{kn}$}
For a $3\times3$ <matrix> $A$, contraction of $\epsilon_{lmn}A_{il}A_{jm}A_{kn}=(\det A)\epsilon_{ijk}$ with $\epsilon_{ijk}$ proves the title identity, because $\epsilon_{ijk}\epsilon_{ijk}=6$. Define
$$
C_{ab}=\frac12\epsilon_{ajk}\epsilon_{bmn}A_{jm}A_{kn}.
$$
Expansion shows that $C$ is the <cofactor matrix>. Equivalently, the determinant identity gives $A_{ib}C_{ab}=(\det A)\delta_{ia}$. Thus for invertible $A$,
$$
(A^{-1})_{ba}=\frac{C_{ab}}{\det A}=\frac{\epsilon_{ajk}\epsilon_{bmn}A_{jm}A_{kn}}{2\det A}.
$$
Differentiating the cubic determinant formula gives three equal terms, so $\partial(\det A)/\partial A_{ab}=C_{ab}$. This is the entrywise <derivative of the determinant> and remains valid for singular $A$ through the cofactor expression. The inverse expression requires $\det A\ne0$.
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