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 expression for \( \text{adj}(\text{adj}(A)) \) where \( A \) is a \( 2 \times 2 \) non-singular matrix.

### Step-by-Step Solution:

**Step 1: Recall the property of adjoint matrices.**
The adjoint of a matrix \( A \) is denoted as \( \text{adj}(A) \). For any square matrix \( A \) of order \( n \), the following property holds:
\[ \text{adj}(A) \cdot A = \det(A) \cdot I_n \]
where \( I_n \) is the identity matrix of order \( n \).

**Hint:** Remember that the adjoint of a matrix is related to its determinant and the identity matrix.

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**Step 2: Apply the property for a \( 2 \times 2 \) matrix.**
For a \( 2 \times 2 \) matrix, we can use the property:
\[ \text{adj}(A) = \det(A) \cdot A^{-1} \]
Thus, we can express the adjoint of the adjoint:
\[ \text{adj}(\text{adj}(A)) = \det(\text{adj}(A)) \cdot A \]

**Hint:** Use the relationship between the adjoint and the determinant to express \( \text{adj}(\text{adj}(A)) \).

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**Step 3: Calculate \( \det(\text{adj}(A)) \).**
For a \( 2 \times 2 \) matrix, the determinant of the adjoint is given by:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
where \( n \) is the order of the matrix. Since \( n = 2 \):
\[ \det(\text{adj}(A)) = \det(A)^{2-1} = \det(A)^1 = \det(A) \]

**Hint:** Remember the formula for the determinant of the adjoint matrix based on the order of the matrix.

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**Step 4: Substitute back into the expression.**
Now substituting back into our expression for \( \text{adj}(\text{adj}(A)) \):
\[ \text{adj}(\text{adj}(A)) = \det(A) \cdot A \]

**Hint:** This step involves substituting the determinant we just calculated into the expression for \( \text{adj}(\text{adj}(A)) \).

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**Step 5: Conclude the result.**
Since \( A \) is a non-singular matrix, \( \det(A) \neq 0 \). Therefore, we can simplify:
\[ \text{adj}(\text{adj}(A)) = A \]

Thus, the final result is:
\[ \text{adj}(\text{adj}(A)) = A \]

### Final Answer:
The correct answer is \( A \).

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