determinant tensorial de ordre 3

det( a^{k}_{ij} ) = ...

... ( a^{1}_{11}a^{2}_{22}a^{3}_{33}+(-1)·a^{1}_{33}a^{2}_{22}a^{3}_{11} )+...

... ( a^{2}_{11}a^{3}_{22}a^{1}_{33}+(-1)·a^{2}_{33}a^{3}_{22}a^{1}_{11} )+...

... ( a^{3}_{11}a^{1}_{22}a^{2}_{33}+(-1)·a^{3}_{33}a^{1}_{22}a^{2}_{11} )+...

... ( a^{1}_{23}a^{2}_{31}a^{3}_{12}+(-1)·a^{3}_{13}a^{2}_{22}a^{1}_{31} )+...

... ( a^{1}_{21}a^{2}_{32}a^{3}_{13}+(-1)·a^{3}_{23}a^{2}_{32}a^{1}_{11} )+...

... ( a^{1}_{31}a^{2}_{12}a^{3}_{23}+(-1)·a^{3}_{33}a^{2}_{12}a^{1}_{21} )+...

... ( a^{1}_{32}a^{2}_{13}a^{3}_{21}+(-1)·a^{3}_{31}a^{2}_{22}a^{1}_{13} )+...

... ( a^{1}_{12}a^{2}_{23}a^{3}_{31}+(-1)·a^{3}_{32}a^{2}_{23}a^{1}_{11} )+...

... ( a^{1}_{13}a^{2}_{21}a^{3}_{32}+(-1)·a^{3}_{33}a^{2}_{21}a^{1}_{12} )

A^{k}_{ij}·a_{i}a_{j} = a^{k}

a_{1} = a & a_{2} = b & a_{3} = c

A^{1}_{11} = (1/a) & A^{2}_{22} = (1/b) & A^{3}_{33} = (1/c)

A^{2}_{11} = (b/a^{2}) & A^{3}_{22} = (c/b^{2}) & A^{1}_{33} = (a/c^{2})

A^{3}_{11} = (c/a^{2}) & A^{1}_{22} = (a/b^{2}) & A^{2}_{33} = (b/c^{2})

A^{1}_{21} = (1/b) & A^{2}_{32} = (1/c) & A^{3}_{13} = (1/a)

A^{1}_{31} = (1/c) & A^{2}_{12} = (1/a) & A^{3}_{23} = (1/b)

A^{1}_{12} = (1/b) & A^{2}_{23} = (1/c) & A^{3}_{31} = (1/a)

A^{1}_{13} = (1/c) & A^{2}_{21} = (1/a) & A^{3}_{32} = (1/b)

A^{1}_{23} = (a/(bc)) & A^{2}_{31} = (b/(ca)) & A^{3}_{12} = (c/(ab))

A^{1}_{32} = (a/(cb)) & A^{2}_{13} = (b/(ac)) & A^{3}_{21} = (c/(ba))

det( A^{k}_{ij} ) = 0