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线性代数中如何证明(A^-1)^-1=A,(AT)^-1=(A^-1)T,(A*)^-1=(A^-1)*,(A*)T=(AT)*
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线性代数中如何证明(A^-1)^-1=A,(AT)^-1=(A^-1)T,(A*)^-1=(A^-
1)*,(A*)T=(AT)*
1)*,(A*)T=(AT)*
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答案和解析
1.因为AA^{-1}=A^{-1}A=E,所以(A^{-1})^{-1}=A;
2.因为A^{T}(A^{-1})^{T}=(A^{-1})^{T}A^{T}=(AA^{-1})^{T}=E^{T}=E,所以(A^{T})^{-1}=(A^{-1})^{T};
3.因为A*=|A|A^{-1},所以(A*)^{-1}=(|A|A^{-1})^{-1}=|A|^{-1}(A^{-1})^{-1}=A/|A|;
(A^{-1})*=|A^{-1}|(A^{-1})^{-1}=A/|A|,
所以(A*)^{-1}=(A^{-1})* ;
4.因为A*=|A|A^{-1},所以
(A*)^{T}=(|A|A^{-1})^{T}=|A|(A^{-1})^{T}=|A|(A^{T})^{-1}=|A^T|(A^{T})^{-1}= (A^{T})*
所以(A*)^{T}=(A^{T})* .
2.因为A^{T}(A^{-1})^{T}=(A^{-1})^{T}A^{T}=(AA^{-1})^{T}=E^{T}=E,所以(A^{T})^{-1}=(A^{-1})^{T};
3.因为A*=|A|A^{-1},所以(A*)^{-1}=(|A|A^{-1})^{-1}=|A|^{-1}(A^{-1})^{-1}=A/|A|;
(A^{-1})*=|A^{-1}|(A^{-1})^{-1}=A/|A|,
所以(A*)^{-1}=(A^{-1})* ;
4.因为A*=|A|A^{-1},所以
(A*)^{T}=(|A|A^{-1})^{T}=|A|(A^{-1})^{T}=|A|(A^{T})^{-1}=|A^T|(A^{T})^{-1}= (A^{T})*
所以(A*)^{T}=(A^{T})* .
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