When working with matrix expressions involving products, it’s essential to understand how transposition interacts with multiplication. In the case of symmetric matrices, the key property \(A^T = A\) and \(B^T = B\) helps simplify complex matrix expressions. By taking the transpose of \(AB - BA\), we can identify whether the result is symmetric or skew-symmetric based on the behavior of its transpose. A result of \( (AB - BA)^T = -(AB - BA) \) directly indicates that the matrix is skew-symmetric.
To ascertain the characteristic of \(AB - BA\), we employ the principles governing symmetric and skew-symmetric matrices.
Symmetric Matrix Principle: A matrix \(M\) is classified as symmetric when its transpose equals itself, i.e., \(M^T = M\).
Given that \(A\) and \(B\) are symmetric matrices, their transposes are equal to themselves: \(A^T = A\) and \(B^T = B\).
Let us now examine the transpose of \((AB - BA)\):
\[ (AB - BA)^T = B^T A^T - A^T B^T \]
Substituting \(A^T = A\) and \(B^T = B\), we get:
\[ (AB - BA)^T = BA - AB = -(AB - BA) \]
This outcome demonstrates that \(AB - BA\) adheres to the definition of a skew-symmetric matrix, given that \((AB - BA)^T = -(AB - BA)\).
Consequently, \(AB - BA\) is identified as skew-symmetric.