Question

If A and B are skew-symmetric matrices of same order, then \( AB' + BA' \) is a/an :

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

Hint:

For any two matrices \( A \) and \( B \), an expression of the form \( XY + YX \) where \( X=A \) and \( Y=B' \) retains its structure under transpose because the reversal law swaps the multiplication order, mirroring the terms back into themselves.

Answer 1

The Correct Option is A

Concept: A square matrix \( M \) is symmetric if \( M' = M \) (where \( M' \) denotes the transpose of \( M \)) and skew-symmetric if \( M' = -M \). Important properties of transposes used here include:
• \( (X + Y)' = X' + Y' \)
• \( (XY)' = Y'X' \)
• \( (X')' = X \)

Step 1: Apply the given conditions for matrices A and B.
Since \( A \) and \( B \) are skew-symmetric matrices, by definition we have: \[ A' = -A \quad \text{and} \quad B' = -B \]

Step 2: Take the transpose of the given matrix expression.
Let the given matrix expression be denoted by \( P \): \[ P = AB' + BA' \] Taking the transpose on both sides: \[ P' = (AB' + BA')' \] Using the sum property of transposes: \[ P' = (AB')' + (BA')' \]

Step 3: Apply the reversal law of transposes.
Using the property \( (XY)' = Y'X' \): \[ P' = (B')'A' + (A')'B' \] Since the transpose of a transpose returns the original matrix (\( (X')' = X \)): \[ P' = BA' + AB' \]

Step 4: Use commutativity of matrix addition to compare with the original matrix.
By rearranging the terms using matrix addition commutativity: \[ P' = AB' + BA' = P \] Since \( P' = P \), the matrix \( AB' + BA' \) is a symmetric matrix. This corresponds to option (A).