If A is a square matrix, then A - A' is a

Options

A. diagonal matrix
B. skew-symmeterix matrix
C. symmeterix matrix
D. None of these

Answer

To solve the problem, we need to analyze the expression \( A - A' \) where \( A' \) denotes the transpose of matrix \( A \).

### Step-by-Step Solution:

1. **Understanding the Transpose**:
- The transpose of a matrix \( A \), denoted as \( A' \), is obtained by flipping the matrix over its diagonal. This means that the element at position \( (i, j) \) in \( A \) becomes the element at position \( (j, i) \) in \( A' \).

2. **Calculate \( A - A' \)**:
- We are interested in the expression \( A - A' \). This means we will subtract the transpose of \( A \) from \( A \).

3. **Transpose of the Difference**:
- We can find the transpose of the expression \( A - A' \):
\[ (A - A')' = A' - (A')' = A' - A \]
- Since \( (A')' = A \), we have:
\[ (A - A')' = A' - A = -(A - A') \]

4. **Conclusion about the Matrix**:
- The result \( (A - A')' = -(A - A') \) indicates that \( A - A' \) is skew-symmetric. A matrix \( B \) is called skew-symmetric if \( B' = -B \).

5. **Final Answer**:
- Therefore, \( A - A' \) is a skew-symmetric matrix.