How to prove that adjoint(AB)= adjoint(B).adjoint(A) if its given that A and B are two square and invertible matrices.
Using formula to calculate inverse of matrix, we can say that
(1)
adj(AB) is adjoint of (AB) and det(AB) is determinant of (AB).
(2)
We also know that
, and
, putting this in above equation (2), we get
(3)
We also know that
and
and
Multiplying the above equations, we can say that
(4)
From equation number (3) and (4), we can say that adj(AB)=adj(B).adj(A) which we wanted to prove.
Nguyen Quynh Phuong says
I am wondering if Det(A)=0 or det(A)=0,does this proof remain accurate?
Nguyen Quynh Phuong says
Sorry, I meant Det(A)=0 or Det(B)=0.
Jashan says
In the top line, I have written that how to prove adj(A.B)=adj(B).adj(A) if A and B are invertible matrices. So, det(A) and det(B) are both non zero for this proof.