Finding the determinant of a matrix given the adjoint 
My attempt:
Knowing that $$A(AdjA) = IdetA$$
I took the determinant on both sides: $$det(A)det(AdjA) = det(det(A))$$
So, $$det(A)det(AdjA) = (det(A))^3$$
$$det(AdjA) = (det(A))^2$$
$$det(A) = (AdjA)^{0.5}$$
The adjoint was calculated to be 4
Thus, $$det(A) = 4^{0.5}=2$$
Why is the answer -2? 

Your computation does not allow you to choose between  $-2$ and $2$. But then you are not using the information you have about  $A $.
Assuming that as you say  $$\tag {1}A (\text {Adj}\,A)=I\,\det A, $$ the $3,3$ entry in (1) is the third row of $A $ times the third column of the adjoint; and this should equal  $\det A $. So
$$
\det A=1\times (-1)+ 2\times (-1)+ 1\times1=-2.
$$
