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https://texercises.com/exercise/determinants-and-transposition/
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Exercise:
forall Ain M_ntimes nK textdetA^TtextdetA.

Solution:
Proof. If sigma is a permutation of degree n then A^Tisigma iAsigma i i. Longrightarrow textdetA^T_sigmatextsgnsigmaAsigma ... Asigma n n. If jsigma i then Asigma i iAj sigma^-j hence Asigma ... Asigma nnAsigma^-... Ansigma^-n because every factor on LHS of will appear exactly once on RHS and vicversa. Now textsgnsigma textsgnsigma^-textsgn id &Longrightarrow textsgnsigma textsgnsigma^-. When sigma runs over mathcalS_n all possible permutation of degree n sigma^- also runs over the entire of mathcalS_n. &Longrightarrow textdetA^T_sigmain mathcalS_ntextsgnsigma Asigma^- Ansigma^-n _tauin mathcalS_ntextsgntauAtau ... Antau n textdetA.
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Exercise:
forall Ain M_ntimes nK textdetA^TtextdetA.

Solution:
Proof. If sigma is a permutation of degree n then A^Tisigma iAsigma i i. Longrightarrow textdetA^T_sigmatextsgnsigmaAsigma ... Asigma n n. If jsigma i then Asigma i iAj sigma^-j hence Asigma ... Asigma nnAsigma^-... Ansigma^-n because every factor on LHS of will appear exactly once on RHS and vicversa. Now textsgnsigma textsgnsigma^-textsgn id &Longrightarrow textsgnsigma textsgnsigma^-. When sigma runs over mathcalS_n all possible permutation of degree n sigma^- also runs over the entire of mathcalS_n. &Longrightarrow textdetA^T_sigmain mathcalS_ntextsgnsigma Asigma^- Ansigma^-n _tauin mathcalS_ntextsgntauAtau ... Antau n textdetA.
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determinant, eth, fs23, lineare algebra, proof
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(3, default)
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Language
ENG (English)
Type
Proof
Creator rk
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