Question

Let A and B be two symmetric matrices of same order. Then, the matrix AB - BA is

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

Hint:

Transpose of difference = Difference of transposes.

Answer 1

The Correct Option is B

Step 1: Properties
A and B are symmetric, so $A^T = A$ and $B^T = B$.
Step 2: Transpose Calculation
Consider $(AB - BA)^T = (AB)^T - (BA)^T$. Using $(XY)^T = Y^T X^T$, we get $B^T A^T - A^T B^T$.
Step 3: Substitution
Substitute $A^T = A$ and $B^T = B$: $BA - AB$.
Step 4: Conclusion
$BA - AB = -(AB - BA)$. Since the transpose is the negative of the original matrix, it is skew-symmetric.
Final Answer: (b)