If A is a square matrix, then what is adj `A^(T) - (adj A)^(T)` equal to ?

Options

A. 2|A|
B. 2|A|I
C. Null Matrix
D. Unit Matrix

Explanation

To solve the problem, we need to find the expression for \( \text{adj}(A^T) - (\text{adj}(A))^T \) where \( A \) is a square matrix.

### Step-by-Step Solution:

1. **Understanding the Adjoint of a Matrix**:
The adjoint (or adjugate) of a square matrix \( A \) is defined as the transpose of its cofactor matrix. For a matrix \( A \), the adjoint is denoted as \( \text{adj}(A) \).

2. **Using the Property of Adjoint**:
A key property of the adjoint is that:
\[ \text{adj}(A^T) = (\text{adj}(A))^T \]
This means that the adjoint of the transpose of a matrix is equal to the transpose of the adjoint of the matrix.

3. **Substituting into the Expression**:
Now, we can substitute this property into our expression:
\[ \text{adj}(A^T) - (\text{adj}(A))^T = (\text{adj}(A))^T - (\text{adj}(A))^T \]

4. **Simplifying the Expression**:
Since both terms are equal, we can simplify the expression:
\[ (\text{adj}(A))^T - (\text{adj}(A))^T = 0 \]

5. **Final Result**:
Therefore, we conclude that:
\[ \text{adj}(A^T) - (\text{adj}(A))^T = 0 \]

### Conclusion:
The final answer is:
\[ \text{adj}(A^T) - (\text{adj}(A))^T = 0 \]