If A is a `2xx2` non singular matrix, then adj(adj A) is equal to :

Options

A. `A^(2)`
B. A
C. `A^(-1)`
D. `(A^(-1))^(2)`

Answer

To solve the problem, we need to find the value of \( \text{adj}(\text{adj}(A)) \) for a \( 2 \times 2 \) non-singular matrix \( A \).

### Step-by-Step Solution:

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

2. **Finding the Determinant of the Adjoint**:
For an \( n \times n \) matrix \( A \), the determinant of the adjoint is given by:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
Since \( A \) is a \( 2 \times 2 \) matrix, we have \( n = 2 \):
\[ \det(\text{adj}(A)) = \det(A)^{2-1} = \det(A) \]

3. **Calculating the Adjoint of the Adjoint**:
Now, we need to find \( \text{adj}(\text{adj}(A)) \). Using the property of determinants:
\[ \det(\text{adj}(\text{adj}(A))) = \det(\text{adj}(A))^{n-1} \]
Again, since \( n = 2 \):
\[ \det(\text{adj}(\text{adj}(A))) = \det(\text{adj}(A))^{2-1} = \det(\text{adj}(A)) = \det(A) \]

4. **Relating Back to the Original Matrix**:
We know that \( \text{adj}(A) \) is a \( 2 \times 2 \) matrix, and we can use the property:
\[ A \cdot \text{adj}(A) = \det(A) I \]
Therefore, we can express:
\[ \text{adj}(\text{adj}(A)) = \det(A) A^{-1} \]

5. **Final Calculation**:
Since \( A \) is non-singular, we can write:
\[ \text{adj}(\text{adj}(A)) = \det(A) A^{-1} \]
For a \( 2 \times 2 \) matrix, it can be shown that:
\[ \text{adj}(\text{adj}(A)) = A \]

### Conclusion:
Thus, we conclude that:
\[ \text{adj}(\text{adj}(A)) = A \]
The correct answer is option B: \( A \).