If A is a square matrix, then what is the simplest way to put Adj (A' ) − (Adj A)'

options given were |A|, 2|A|, Null and unit matrices...
options given were |A|, 2|A|, Null and unit matrices...
Step 1: Note down the given equations and recall the definitions which is required
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)