Question

show that (adjAB)= (adjB) (adjA) whereA and B are two non singular matrics of same order n

Answer 1

We have for any square matrix A the following
AadjA=detA⋅I
Therefore for two square matrices A,B of the same order we get
AB⋅adjAB=det(AB)⋅I=detAdetB⋅I=detBdetA⋅I=detB⋅AadjA
But
detB⋅AadjA=A⋅detB⋅adjA=A⋅(detB⋅I)⋅adjA=AB⋅adjBadjA
Thus we obtain
AB⋅adjAB=AB⋅adjBadjA
If AB is invertible then multiplying both sides by (AB)−1 yields
adjAB=adjBadjA
If (AB) is not invertible then we can at most say
AB⋅(adjAB−adjBadjA)=0