Question

For any square matrix A, A - A\(^T\) is always:

• A. A null matrix
• B. A symmetric matrix
• C. A skew-symmetric matrix
• D. An identity matrix

Hint:

Any square matrix \( A \) can be represented as the sum of a symmetric and a skew-symmetric matrix: \( A = \frac{A+A^T}{2} + \frac{A-A^T}{2} \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A matrix \( S \) is skew-symmetric if \( S^T = -S \). We need to check this property for \( S = A - A^T \).

Step 2: Detailed Explanation:
Let \( S = A - A^T \). Take the transpose of \( S \):
\[ S^T = (A - A^T)^T = A^T - (A^T)^T \]
Using the property \((A^T)^T = A\):
\[ S^T = A^T - A \]
Factor out the negative sign:
\[ S^T = -(A - A^T) = -S \]
Since \( S^T = -S \), the matrix \( A - A^T \) is skew-symmetric.

Step 3: Final Answer:
The matrix is a skew-symmetric matrix.