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    Member Jskid's Avatar
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    theoretical question involving determinant and adjoint

    Show that if A is an nxn matrix then det(adj A)=[det(A)]^{n-1}
    I'm not good with adjoints...first thing that comes to mind is A(adj A)=(adj A)A=det(A)I_n
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jskid View Post
    Show that if A is an nxn matrix then det(adj A)=[det(A)]^{n-1}
    I'm not good with adjoints...first thing that comes to mind is A(adj A)=(adj A)A=det(A)I_n
    Clearly you mean adjugate, and not adjoint. Use precisely the property you claimed. Namely, \displaystyle \text{adj}(A)A=\det(A)I_n. Taking the determinants of both sides gives \det(\text{adj}(A))\det(A)=\det(\text{adj}(A)A)=\d  et(\det(A)I_n)=\det^n(A)\det(I_n)=\det(A)^n.
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    Member Jskid's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Clearly you mean adjugate, and not adjoint.
    My text book calls it adjoint. What's the difference?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jskid View Post
    My text book calls it adjoint. What's the difference?
    What you are calling the adjoint is what I call the adjugate. The adjoint of an endomorphism A:V\to V (if it exists) on an inner product space \left(V,\langle\cdot,\cdot\rangle\right) is the unique endomorphism A^\ast:V\to V such that \left\langle A(x),y\right\rangle=\left\langle x,A^\ast(y)\right\rangle for every x,y\in V. It has the property that if you fix an orthonormal ordered basis (assuming \dim V<\infty) \mathcal{B} then \left[A^\ast\right]_B=A\right]_B^\ast (where here [\cdot]_B is the matrix representation with respect to that ordered basis, and [A]_B^\ast=\left(\overline{[A]_B}\right)^{\top}).
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    Quote Originally Posted by Jskid View Post
    My text book calls it adjoint. What's the difference?
    My linear book uses the term adjoint to so now worries.
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    Re: theoretical question involving determinant and adjoint

    My linear book describes adjoint as the linear operator T* such that (T*(v), w) = (v, T(w)). * relates to linear operators in much the same way as * relates to matrices.

    Of course, this isn't the first time mathematicians use the same word to describe multiple mathematical concepts.
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    Forum Admin topsquark's Avatar
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    Re: theoretical question involving determinant and adjoint

    Quote Originally Posted by mathguy25 View Post
    My linear book describes adjoint as the linear operator T* such that (T*(v), w) = (v, T(w)). * relates to linear operators in much the same way as * relates to matrices.

    Of course, this isn't the first time mathematicians use the same word to describe multiple mathematical concepts.
    Ummm...It's occasionally useful to add a post on older material, but you are aware you just helped out on a post that is more than 2 years old?

    -Dan
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