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Math Help - Linear algebra

  1. #1
    Member kezman's Avatar
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    Linear algebra

    Let  A \in M_n(k) i
    1 A is Invertible. Calculate Det(adj A)
    2 If A is antisimetric and n is odd, Prove that Det = 0
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  2. #2
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    Quote Originally Posted by kezman View Post
    Let  A \in M_n(k) invertible matrix.
    1 Calculate Det(adj A)
    2 If A is antisimetric and n is odd, Prove that Det = 0
    If A is an invertible matrix. Then, A^{-1} = \frac{1}{\det (A)}\mbox{adj}(A). Thus, \det (A)A^{-1} = \mbox{adj}(A). Thus, \det (\det(A) A^{-1} ) = \det (\mbox{adj}(A)). Thus, [\det(A)]^k \cdot \frac{1}{\det (A)} = \det (\mbox{adj}(A)).
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  3. #3
    MHF Contributor red_dog's Avatar
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    2) If A is antisimetric then A=-A^T
    Then \det A=\det(-A^T)=(-1)^n\det A^T=-\det A\Rightarrow \det A=0
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  4. #4
    Member kezman's Avatar
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    Thank you very much.
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