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Need help to proof Adjugate Matrix

  1. Aug 22, 2007 #1
    Alo! I kinda need some assistance to proof this:

    "Show that adj(adj A) = |A|^(n-2). A, if A is a (n x n) square matrix and |A| is not equal to zero"

    NOTE: 1) adj(A) = adjugate of matrix A,
    2) |A| = determinant of A,
    3) ^ = power

    I've tried to work around the equation using the formula: A^-1 = |A|^-1. adj(A), BUT doesn't seem to work at all. Sooo HELP!!!..and thanks in advance :biggrin:.
     
  2. jcsd
  3. Aug 22, 2007 #2

    matt grime

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    The definition of Adj(A) is *not* given by a formula involving A inverse. It is defined even if A is not invertible.

    Adj(X) satisfies X*Adj(X)=det(X)I.
     
  4. Aug 22, 2007 #3
    We have [tex]A\cdot A^*=\det A\cdot I_n[/tex] (1)
    Then [tex]det A\cdot\det A^*=(\det A)^n\Rightarrow\det A^*=(\det A)^{n-1}[/tex]
    Applying (1) for [tex]A^*[/tex] we have
    [tex]A^*\cdot (A^*)^*=\det A^*\cdot I_n[/tex].
    Multiply both members by [tex]A[/tex]
    [tex]A\cdot A^*(A^*)^*=det A^*\cdot A\Rightarrow \det A\cdot (A^*)^*=(\det A)^{n-1}\cdot A\Rightarrow[/tex]
    [tex]\Rightarrow (A^*)^*=(det A)^{n-2}\cdot A[/tex]
     
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