How can one prove the following identity of the cross product? ( M a ) &#x00D7;<!--

ntaraxq 2022-07-12 Answered
How can one prove the following identity of the cross product?
( M a ) × ( M b ) = det ( M ) ( M T ) 1 ( a × b )
a and b are 3-vectors, and M is an invertible real 3 × 3 matrix.
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Answers (1)

Tatiana Gentry
Answered 2022-07-13 Author has 10 answers
Recall that u × v is the only vector w R 3 that satisfies x R 3 , det ( u , v , x ) = w , x So, we have det ( M ) a × b , x = det ( M ) det ( a , b , x ) = det ( M a , M b , M x ) = M a × M b , M x = M T ( M a × M b ) , x Thus x R 3 , det ( M ) ( a × b ) , x = M T ( M a × M b ) , x It follows that det ( M ) ( a × b ) = M T ( M a × M b ) or equivalently, for an invertible matrix M, det ( M ) ( M T ) 1 ( a × b ) = M a × M bRemark. The most general formula is C o ( M ) ( a × b ) = M a × M bwhere C o ( M ) is the matrix of cofactors of M. Here we do not need M to be invertible. This follows from the proved formula by continuity.
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