# Thread: theoretical question involving determinant and adjoint

1. ## theoretical question involving determinant and adjoint

Show that if A is an nxn matrix then $\displaystyle det(adj A)=[det(A)]^{n-1}$
I'm not good with adjoints...first thing that comes to mind is $\displaystyle A(adj A)=(adj A)A=det(A)I_n$

2. Originally Posted by Jskid Show that if A is an nxn matrix then $\displaystyle det(adj A)=[det(A)]^{n-1}$
I'm not good with adjoints...first thing that comes to mind is $\displaystyle A(adj A)=(adj A)A=det(A)I_n$
Clearly you mean adjugate, and not adjoint. Use precisely the property you claimed. Namely, $\displaystyle \displaystyle \text{adj}(A)A=\det(A)I_n$. Taking the determinants of both sides gives $\displaystyle \det(\text{adj}(A))\det(A)=\det(\text{adj}(A)A)=\d et(\det(A)I_n)=\det^n(A)\det(I_n)=\det(A)^n$.

3. Originally Posted by Drexel28 Clearly you mean adjugate, and not adjoint.
My text book calls it adjoint. What's the difference?

4. Originally Posted by Jskid 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 $\displaystyle A:V\to V$ (if it exists) on an inner product space $\displaystyle \left(V,\langle\cdot,\cdot\rangle\right)$ is the unique endomorphism $\displaystyle A^\ast:V\to V$ such that $\displaystyle \left\langle A(x),y\right\rangle=\left\langle x,A^\ast(y)\right\rangle$ for every $\displaystyle x,y\in V$. It has the property that if you fix an orthonormal ordered basis (assuming $\displaystyle \dim V<\infty$) $\displaystyle \mathcal{B}$ then $\displaystyle \left[A^\ast\right]_B=A\right]_B^\ast$ (where here $\displaystyle [\cdot]_B$ is the matrix representation with respect to that ordered basis, and $\displaystyle [A]_B^\ast=\left(\overline{[A]_B}\right)^{\top}$).

5. Originally Posted by Jskid My text book calls it adjoint. What's the difference?
My linear book uses the term adjoint to so now worries.

6. ## 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.

7. ## Re: theoretical question involving determinant and adjoint Originally Posted by mathguy25 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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