SOLUTION: If A inverse = (see link), find det(Adj A) Link: https://ibb.co/GP9XwKX

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Question 1199086: If A inverse = (see link), find det(Adj A)
Link: https://ibb.co/GP9XwKX

Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.

                Let solve it step by step. 


(1)  First calculate determinant of the given matrix A%5E%28-1%29.

     Use the basic definition of the determinant in an expanded form.

     You will get  det%28A%5E%28-1%29%29 = 1%2F3.




(2)  It implies, due to well known properties of determinants, that

        det(A) = %281%2F3%29%5E%28-1%29 = 3.



(3)  Finally, use another well known formula of linear algebra, which for 3x3 matrix A takes the form

        det(Adj(A)) = %28det%28A%29%29%5E2.


     We just know from  (2)  that det(A) = 3.  Therefore,

        det(Adj(A)) = 3%5E2 = 9.


ANSWER.  det(Adj(A)) = 9.

Solved, with all necessary explanations.