We start with the well-known formula for any invertible matrix A : A⋅adj(A)=|A|I Since A is non-singular (|A|≠0), we can write adj(A)=|A|A−1 Now, replace A with A−1. Then, applying the same formula for A−1, we have: A−1⋅adj(A−1)=|A−1|I Recall that |A−1|=‌
1
|A|
. Thus, A−1⋅adj(A−1)=‌
1
|A|
I Multiplying both sides on the left by A (which is valid since A is invertible) gives: A⋅A−1⋅adj(A−1)=A⋅‌
1
|A|
I Since A⋅A−1=I, this simplifies to: adj(A−1)=‌
1
|A|
A. On the other hand, we earlier had: adj(A)=|A|A−1
Taking the inverse of both sides (and recalling that for any nonzero scalar c and invertible matrix B,(cB)−1=‌
1
c
B−1 ), we get: (adj(A))−1=(|A|A−1)−1=‌
1
|A|
(A−1)−1=‌
1
|A|
A. Thus, we obtain: adj(A−1)=(adj(A))−1 Therefore, the correct option is: Option A - (adj‌A)−1.