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 of finding \( \text{adj}(\text{adj}(A)) \) for a \( 2 \times 2 \) non-singular matrix \( A \), we can follow these steps:

### Step 1: Understand the properties of the adjoint
The adjoint 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 and \( \det(A) \) is the determinant of \( A \).

### Step 2: Apply the property of the adjoint
For a \( 2 \times 2 \) matrix, we have the property:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
where \( n \) is the order of the matrix. Here, \( n = 2 \), so:
\[ \det(\text{adj}(A)) = \det(A)^{2-1} = \det(A) \]

### Step 3: Find \( \text{adj}(\text{adj}(A)) \)
Using the same property for the adjoint again:
\[ \det(\text{adj}(\text{adj}(A))) = \det(\text{adj}(A))^{2-1} = \det(\text{adj}(A)) = \det(A) \]

### Step 4: Relate back to the original matrix
From the properties of determinants and adjoints, we know:
\[ \text{adj}(\text{adj}(A)) = \det(A) A^{-1} \]
For a non-singular matrix \( A \), \( A^{-1} \) exists.

### Step 5: Conclude the result
Thus, we can express \( \text{adj}(\text{adj}(A)) \) as:
\[ \text{adj}(\text{adj}(A)) = \det(A) A^{-1} \]
For a \( 2 \times 2 \) matrix, this simplifies to:
\[ \text{adj}(\text{adj}(A)) = \det(A) A^{-1} \]
Since \( \det(A) \) is a scalar, we can denote it as \( k \) (where \( k = \det(A) \)), leading to:
\[ \text{adj}(\text{adj}(A)) = k A^{-1} \]
However, since \( A \) is a non-singular matrix, we can conclude that:
\[ \text{adj}(\text{adj}(A)) = A^2 \]
Thus, the answer is:
\[ \text{adj}(\text{adj}(A)) = A^2 \]

### Final Answer
The correct option is \( A^2 \).

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