Question

If \(A\) and \(B\) are two skew symmetric matrices of the same order then \(AB\) is skew symmetric if and only if

• A. \(AB+BA=0\)
• B. \(AB-BA=0\)
• C. \(AB+BA=1\)
• D. \(AB-BA=1\)

Hint:

For skew symmetric matrices, always start with: \[ A^T=-A \] and use: \[ (AB)^T=B^TA^T \]

Answer 1

The Correct Option is A

Concept:
A matrix \(A\) is called skew symmetric if: \[ A^T=-A \] Similarly, if \(B\) is skew symmetric: \[ B^T=-B \] We need to find the condition under which \(AB\) is also skew symmetric. For \(AB\) to be skew symmetric: \[ (AB)^T=-AB \]

Step 1: Use transpose of product rule.
We know: \[ (AB)^T=B^TA^T \] Since \(A\) and \(B\) are skew symmetric: \[ A^T=-A \] and \[ B^T=-B \] Therefore: \[ (AB)^T=(-B)(-A) \] \[ (AB)^T=BA \]

Step 2: Apply skew symmetric condition on \(AB\).
For \(AB\) to be skew symmetric: \[ (AB)^T=-AB \] But we found: \[ (AB)^T=BA \] So: \[ BA=-AB \]

Step 3: Rearrange the equation.
\[ BA=-AB \] Bring all terms to one side: \[ AB+BA=0 \] This is the required condition.

Step 4: Check the options.
Option (A) \(AB+BA=0\) is correct.
Option (B) \(AB-BA=0\) means \(AB=BA\), which would generally make \(AB\) symmetric, not skew symmetric.
Option (C) and option (D) are invalid because \(1\) is not the correct zero matrix condition. Hence, the correct answer is: \[ \boxed{(A)\ AB+BA=0} \]