Question

If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?

• A. \(A^2 + B^2\)
• B. \(A^2 - B^2\)
• C. \(AB + BA\)
• D. \(AB - BA\)

Hint:

Take the transpose using (XY)^T = Y^T X^T with A^T = A and B^T = B. Only AB - BA changes sign.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A matrix \(M\) is symmetric if \(M^T = M\), and skew-symmetric if \(M^T = -M\).
A is symmetric, so \(A^T = A\). Likewise \(B^T = B\).

Step 2: Key rule:
For matrices of the same order, \((XY)^T = Y^T X^T\).
So \((AB)^T = B^T A^T = BA\) and \((BA)^T = A^T B^T = AB\).

Step 3: Check option (D) \(AB - BA\):
\[ (AB - BA)^T = (AB)^T - (BA)^T = BA - AB = -(AB - BA) \]
The transpose equals the negative of the matrix, so \(AB - BA\) is skew-symmetric.

Step 4: Check option (C) \(AB + BA\):
\[ (AB + BA)^T = BA + AB = AB + BA \]
This equals the matrix itself, so it is symmetric, not skew-symmetric.

Step 5: Check options (A) and (B):
\((A^2)^T = (A^T)^2 = A^2\) and \((B^2)^T = B^2\). So \(A^2\) and \(B^2\) are symmetric.
Their sum and difference are symmetric too. So (A) and (B) are symmetric, and are not skew-symmetric in general.

Final Answer:
The skew-symmetric matrix is \(AB - BA\), option (D). \[ \boxed{AB - BA} \]