Binet-Cauchy Identity

Author

Eric W. Weisstein
October 10, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Binet-CauchyIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Binet-CauchyIdentity.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Check the Identity

BinetCauchyIdentity[n_]:=Module[{aa=Array[a,n],bb=Array[b,n],cc=Array[c,n],dd=Array[d,n]},
aa.cc bb.dd-aa.dd bb.cc==
Plus@@((a[#1]b[#2]-a[#2]b[#1])(c[#1]d[#2]-c[#2]d[#1])&@@@Subsets[Range[n],{2}])
]

Expand//@Table[BinetCauchyIdentity[n],{n,10}]

Table[BinetCauchyIdentity[n]/.a_[k_]:>Subscript[a,k],{n,3}]

Cross[Array[a,3],Array[b,3]].Cross[Array[c,3],Array[d,3]]/.a_[k_]:>Subscript[a,k]

Degenerates to Langrange's Identity

Table[BinetCauchyId[n] /. {c->a,d->b}/.a_[k_]:>Subscript[a,k],{n,2,3}]