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Math Help - Determinant Proof

  1. #1
    Newbie kaylakutie's Avatar
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    Determinant Proof

    Stuck on two homework problems.

    1. Let A and B be n x n matricies such that AB = I. Prove that det(A) is not equal to 0 and det(B) is not equal to 0.

    2. Let A and B be n x n matricies such that AB is singular. Prove that either A or B is singular.
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  2. #2
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    1.
    AB=I
    det(AB)=det(I)
    det(A)det(B)=det(I)
    det(I)=1.
    If det(A) v det(B)=0, then you get zero times another scalar, which is zero.
    Hence, det(A) and det(B) ≠ 0.

    QED

    2.

    If AB is singular, then,

    det(AB)=0.
    So, det(A)det(B)=0.
    If neither scalars det(A) or det(B)≠0, then neither one of them could be singular, sinceall nonsingular matricies have nonzero determinants; so, det(A)det(B) wouldn't be equal to zero.
    So, A is singular, or B is singular.

    QED

    Recall that for all scalars a b in R ab=0 iff (a=0)v(b=0)
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