Let A be a `2xx2` matrix.
Assertion (A) : `adj(adjA)=A`
Reason (R ) : `|adjA|=|A|`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to analyze the assertion and reason given for a 2x2 matrix A.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that \( \text{adj}(\text{adj} A) = A \). We need to verify if this is true for a 2x2 matrix.

2. **Using the Formula for Adjoint**:
The formula for the adjoint of a matrix \( A \) of order \( n \) is given by:
\[ \text{adj}(\text{adj} A) = |A|^{n-2} A \]
For a 2x2 matrix, \( n = 2 \).

3. **Calculating \( \text{adj}(\text{adj} A) \)**:
Substituting \( n = 2 \) into the formula:
\[ \text{adj}(\text{adj} A) = |A|^{2-2} A = |A|^0 A = 1 \cdot A = A \]
Thus, the assertion \( \text{adj}(\text{adj} A) = A \) is true.

4. **Understanding the Reason**:
The reason states that \( |\text{adj} A| = |A| \). We need to verify if this is correct.

5. **Using the Determinant Property**:
For a 2x2 matrix \( A \), the determinant of the adjoint is given by:
\[ |\text{adj} A| = |A|^{n-1} \]
For \( n = 2 \):
\[ |\text{adj} A| = |A|^{2-1} = |A|^1 = |A| \]
Therefore, the reason \( |\text{adj} A| = |A| \) is also true.

6. **Conclusion**:
Both the assertion and the reason are correct. Thus, the final conclusion is that both the assertion and the reason are true, but the explanation provided may not be entirely clear.