Step 1: Note down the given equations and recall the definitions which is required

•   A is a square matrix (n*n matrix)
• The adjoint of a square matrix A = [aij]n x n is defined as the transpose of the matrix [Aij]n x n, where Aij is the cofactor of the element aij. Adjoing of the matrix A is denoted by adj A.

• Transpose of a Matrix ( A^T)  . If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. 

Step 2:  Find the relation between the adjoint and inverse of a matrix

            NOTE:  A^{-1}=\frac{\text{adj}\ A}{\det A}

Step 3: Replace A by the A^T.

             (A^T)^{-1} = \frac{adj A^T}{det A^T}

               \det A^T*(A^T)^{-1}=\text{adj}\ A^T

Step 4:  We know that det A^T = det A and (A^T)^{-1} = (A^{-1})^T. Use this to simplify

          \text{adj}\ A^T=\det A^T*(A^T)^{-1}

          \text{adj}\ A^T=\det A*(A^{-1})^T

          \text{adj}\ A^T=(\det A*A^{-1})^T.............(1)\ \ \ \ \ (Since\ K*A^T=(KA)^T

Step 5: Use the relation between the adjoint and inverse of a matrix

            NOTE:  we know   A^{-1} = \frac{adj A}{det A}

                    \text{adj}\ A=A^{-1}\det A  ...........(2)

Step 6: Substitute equation (2) in (1)

            We can write \text{adj}\ (A^T)=(\text{adj}\ A)^T

            \text{adj}\ (A^T)-(\text{adj}\ A)^T=0(null)