how do i find det(adj A) ?

where A is an nxn matrix.

thanks xxxxxxxx
You know that A adj(A) = det(A) I, so take determinants:

(I used det(AB)=det(A)det(B) and det(cA)=c^ndet(A).)

So if A is invertible then det(adj(A)) = (det(A))^(n-1). Otherwise det(adj(A))=0. Hence in both cases det(adj(A)) = (det(A))^(n-1).
dvs
You know that A adj(A) = det(A) I, so take determinants:

(I used det(AB)=det(A)det(B) and det(cA)=c^ndet(A).)

So if A is invertible then det(adj(A)) = (det(A))^(n-1). Otherwise det(adj(A))=0. Hence in both cases det(adj(A)) = (det(A))^(n-1).

thanks that is helpful. one thing, where did you get the c^n from?
If you multiply each of the n rows by c, then you multiply the determinant by c n times.
dvs
If you multiply each of the n rows by c, then you multiply the determinant by c n times.

ok. but where does the c come from?!
Oh,
det(det(A) I) = (det(A))^n det(I)

And det(I) = 1.

Take determinant of both sides

Simplify

So as you can see, you just need to find det(A) in order to know det(adj(A))
(edited 9 years ago)
Original post by TeslerCoil

Take determinant of both sides

Simplify