Question

If \( A \) and \( B \) are square matrices of the same order, then \( (A B^T - B A^T) \) is a:

• A. Symmetric matrix
• B. Skew-symmetric matrix
• C. Null matrix
• D. Unit matrix

Hint:

To check if a matrix is skew-symmetric, verify if its transpose is equal to the negative of the original matrix.

Answer 1

The Correct Option is B

To solve the problem, we need to analyze the nature of the matrix expression \( AB^T - BA^T \) when \( A \) and \( B \) are square matrices of the same order.

1. Take the Transpose of the Expression:
Let’s compute the transpose of \( AB^T - BA^T \):

\( (AB^T - BA^T)^T = (AB^T)^T - (BA^T)^T \)
Using the identity \( (XY)^T = Y^T X^T \), we get:

\( (AB^T)^T = B A^T \), and \( (BA^T)^T = A B^T \)
So:
\( (AB^T - BA^T)^T = B A^T - A B^T = - (AB^T - BA^T) \)

2. Interpretation:
Since the transpose of the expression equals the negative of the expression, it satisfies the condition for a skew-symmetric matrix:

\( M^T = -M \Rightarrow M \) is skew-symmetric

3. Conclusion:
The matrix \( AB^T - BA^T \) is skew-symmetric.

Final Answer:
The correct option is (B) skew-symmetric matrix.