Adjugate invertible matrix

Question

If A is an invertible $nxn$ matrix prove that:$ adj(adjA)=(A)(detA)^{n-2}$ I have done this but it somewhere went wrong: $ adj(adjA)=adj(A^{-1} detA)=(A^{-1}detA)^{-1} det(A^{-1}detA)=AdetA det(A^{-1}detA)= Adet(AA^{-1}detA)=A (detA)^n $

Answer 1

For all invertible matrices $A$ and $\lambda\neq0,$ $(\lambda A)^{-1}=\frac1\lambda A^{-1}$.

Answer 2

There are errors in what you have typed. Let me point out the first one.

$ \begin{align} adj(adjA) & =adj(A^{-1} detA)\\ & =(A^{-1}detA)^{-1} det(A^{-1}detA)\\ & =AdetA det(A^{-1}detA) \\ \end{align} $

It should be $ \frac{1}{\det A } $.

Fix that first.