Given a matrix $A$ which is invertible and with determinant>0, then $ \operatorname{adj} A$ is the matrix such that:
$$ (\operatorname{adj} A) A = A (\operatorname{adj} A) = (\det A) I \tag{1}$$
The construction of $\operatorname{adj} A$ is that we take the cofactor matrix then it's transpose. How would we show that the construction would be the one that leads to property (1) intuitively?