Let $$A$$ be any $$3 \times 3$$ invertible matrix. Then which one of the following is not always true?

For an invertible square matrix we always begin with the basic relation

$$A \; adj(A)=adj(A)\;A=|A|\,I,$$

where $$|A|$$ is the determinant of $$A$$ and $$I$$ is the identity matrix of the same order. This identity is true for every non-singular matrix and will be used repeatedly.

Taking determinant on both sides of $$A\,adj(A)=|A|\,I$$ we have

$$|A\,adj(A)|=||A|\,I|.$$

The left-hand side gives $$|A|\,|adj(A)|$$ while the right-hand side equals $$|A|^{n}$$ because the determinant of $$|A|I$$ (an $$n\times n$$ scalar matrix) is $$|A|^{n}$$. Hence

$$|A|\,|adj(A)|=|A|^{n}\;\;\Longrightarrow\;\;|adj(A)|=|A|^{\,n-1}.$$

In the present problem $$n=3$$, so

$$|adj(A)|=|A|^{\,2}.\qquad(1)$$

Next, from $$A\,adj(A)=|A|\,I$$ we isolate $$adj(A)$$:

$$adj(A)=|A|\,A^{-1}.\qquad(2)$$

Because $$A$$ is invertible, the matrix $$adj(A)$$ is also invertible, and its inverse is obtained directly from (2):

$$(adj(A))^{-1}=\bigl(|A|\,A^{-1}\bigr)^{-1}=\dfrac{1}{|A|}\,A.\qquad(3)$$

Now we wish to express $$adj(adj(A))$$. For any non-singular matrix $$B$$ we have the key identity

$$B\,adj(B)=|B|\,I.$$

We apply it with $$B=adj(A)$$:

$$adj(A)\;adj\!\bigl(adj(A)\bigr)=|adj(A)|\,I.\qquad(4)$$

Using (1) to replace $$|adj(A)|$$ inside (4) gives

$$adj(A)\;adj\!\bigl(adj(A)\bigr)=|A|^{2}\,I.\qquad(5)$$

Now we substitute $$adj(A)=|A|\,A^{-1}$$ from (2) into (5):

$$|A|\,A^{-1}\;adj\!\bigl(adj(A)\bigr)=|A|^{2}\,I.$$

Dividing by $$|A|$$ on both sides, we obtain

$$A^{-1}\;adj\!\bigl(adj(A)\bigr)=|A|\,I.$$

Multiplying on the left by $$A$$ gives the desired explicit expression:

$$adj\!\bigl(adj(A)\bigr)=|A|\,A.\qquad(6)$$

Equation (6) is crucial for comparing with the four given statements.

Let us examine every option one by one.

Option A: The right-hand side is

$$|A|^{2}\,(adj(A))^{-1}=|A|^{2}\left(\dfrac{1}{|A|}A\right)=|A|\,A,$$

which is exactly the left-hand side $$adj(adj(A))$$ from (6). So Option A is always true.

Option B: The right-hand side is

$$|A|\,(adj(A))^{-1}=|A|\left(\dfrac{1}{|A|}A\right)=A,$$

whereas (6) tells us that $$adj(adj(A))=|A|\,A$$. These two matrices are equal only in the special case $$|A|=1$$; in general they are different. Hence Option B is not always true.

Option C: This is exactly equation (6). Therefore it is always true.

Option D: This is formula (2), a standard identity valid for every invertible matrix. Hence it is always true.

Only Option B fails to hold for an arbitrary invertible $$3\times3$$ matrix.

Hence, the correct answer is Option B.