Let A be a `2xx2` matrix.
Statement 1: `adj(adjA)=A`
Statement 2: `|adjA|=|A|`.
Which statement is true

Options

A. Statement 1 is right
B. Statement 2 is right
C. Both statement are right and statement 2 explain statement 1
D. Both statement are right and statement 2 does not explain statement 1

Answer

To solve the problem, we need to evaluate the two statements regarding the adjoint of a 2x2 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 as the transpose of the cofactor matrix of \( A \).

2. **Statement 1: \( \text{adj}(\text{adj}(A)) = A \)**:
- For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the adjoint is calculated as:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
- Now, we need to find \( \text{adj}(\text{adj}(A)) \):
\[ \text{adj}(\text{adj}(A)) = \text{adj}\left(\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\right) = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = A \]
- Therefore, Statement 1 is **true**.

3. **Statement 2: \( |\text{adj}(A)| = |A| \)**:
- The determinant of a 2x2 matrix \( A \) is given by:
\[ |A| = ad - bc \]
- The determinant of the adjoint of \( A \) for a 2x2 matrix is given by:
\[ |\text{adj}(A)| = |A|^1 = |A| \]
- Therefore, Statement 2 is also **true**.

4. **Conclusion**:
Both statements are true. Thus, the answer is that both Statement 1 and Statement 2 are correct.