If `A=[(a,b),(c,d)]`, then `adj(adjA)` is equal to

Options

A. `A`
B. `A'`
C. `adjA`
D. `-A`

Explanation

To find \( \text{adj}(\text{adj} A) \) for the matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), we will follow these steps:

### Step 1: Find the adjoint of matrix \( A \)

The adjoint of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

### Step 2: Find the adjoint of \( \text{adj}(A) \)

Now we need to find the adjoint of the adjoint matrix we just calculated:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Using the same formula for the adjoint, we have:

\[ \text{adj}(\text{adj}(A)) = \begin{pmatrix} a & -(-b) \\ -(-c) & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

### Step 3: Conclusion

Thus, we find that:

\[ \text{adj}(\text{adj}(A)) = A \]

### Final Answer

Therefore, \( \text{adj}(\text{adj} A) = A \).

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