Determinant of any square matrix is equal to determinant of its transpose. Lets take an example of any square matrix and find value of its determinant. Then transpose this matrix and again find value of determinant of transpose of matrix. We will note that determinant of matrix is equal to determinant of its transpose.
How to prove that adj(AB)=adj(B).adj(A)?
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)
How to prove that A.adj(A)= adj(A).A=det(A).I ?
Note: This property holds for square matrices.
If, we are given matrix A then
How to prove that
? where adj(A) is adjoint of A and det(A) is determinant of A.
How to prove that adj(adjA) =A.(det(A)) power n-2 ?
How to prove that det(adj(A))= (det(A)) power n-1?
Note: This property holds for square matrices which are invertible.
This property of adjoint of matrices can be easily proved using property
where adj(A) is adjoint of A, det(A) is determinant of A and
is inverse of A. A here is an invertible matrix.