Let A be any `3xx3` invertible matrix. Thenwhich one of the following is not always true?

Options

A. `adj(adj (A))=|A|(adj(A))^(-1)`
B. `adj(adj(A))=|A|^2. (adj(A))^(-1)`
C. `adj(A)=|A|A
D. `adj(adj(A))=|A|.A^(-1)'

Answer

To solve the problem, we need to analyze the properties of the adjoint of a matrix and how they relate to the determinant of an invertible matrix \( A \).

### Step-by-Step Solution:

1. **Understanding the Adjoint Matrix**:
The adjoint of a matrix \( A \), denoted as \( \text{adj}(A) \), is defined such that:
\[ A \cdot \text{adj}(A) = \det(A) I_n \]
where \( I_n \) is the identity matrix of the same order as \( A \).

2. **Using the Property of Adjoint**:
For a \( 3 \times 3 \) matrix \( A \):
\[ A \cdot \text{adj}(A) = \det(A) I_3 \]
This property is always true for any invertible matrix \( A \).

3. **Adjoint of the Adjoint**:
There is a property that states:
\[ \text{adj}(\text{adj}(A)) = \det(A)^{n-2} A \]
For \( n = 3 \):
\[ \text{adj}(\text{adj}(A)) = \det(A)^{3-2} A = \det(A) A \]
This means that this property is also true.

4. **Inverse of the Adjoint**:
We also have:
\[ \text{adj}(A) = \det(A) A^{-1} \]
This is true for any invertible matrix \( A \).

5. **Determinant of the Adjoint**:
The determinant of the adjoint can be expressed as:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
For \( n = 3 \):
\[ \det(\text{adj}(A)) = \det(A)^2 \]
This property holds true as well.

6. **Identifying the False Statement**:
Now, we need to determine which of the given statements is not always true. Based on the properties discussed:
- The first statement is likely the one that does not hold true universally among the options provided.

### Conclusion:
After evaluating the properties of the adjoint and the determinant, we conclude that the first option is the one that is not always true.