Determinant Multiplication

Question

I know the following property of the Determinant:
$Det(A\cdot B)=Det(A)\cdot Det(B)$
When Trying to prove $Det(Adj(A))=Det(A)^{n-1}$ I came across the following dilemma:
$A\cdot Adj(A)=Det(A)\cdot I$ (multipling both sides in Det)
$Det(A)\cdot Det(Adj(A))=Det(Det(A)\cdot I)$
Does $Det(Det(A)\cdot I)=Det(Det(A))\cdot Det(I)$?
Maybe the problem is that $Det(Det(A)\cdot I)$ can not be written as $Det(Det(A))\cdot Det(I)$ as the $Det(A)$ is scalar?

Answer 1

Call $a= \det A$. Then $aI$ is a diagonal matrix, so its determinant is the product of its diagonal entries. So $$\det( (\det A)I) = \det(aI) = a^n = (\det A)^n$$