Matrices and Determinants

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.