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Math Help - Adjoint

  1. #1
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    Adjoint

    I need to prove that for a square matrix (nXn) A: if rkA= n-1 then rk adjA= 1.
    Thanks for any help...
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  2. #2
    MHF Contributor

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    The adjoint of a matrix, A, can be defined as the matrix such that (Au)\cdot v= u \cdot(adj A v).

    If A has rank n- 1, then it had "nullity" n- (n-1)= 1. That is, the set of all vectors u, such that Au= 0, is a one dimensional subspace. In particular, for any such u, (Au)\cdot v= 0= u\cdot(adj A v).
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