Question

If $A$ and $B$ are symmetric matrices of the same order, then $AB - BA$ is:

• A. Symmetric matrix
• B. Zero matrix
• C. Skew-symmetric matrix
• D. Identity matrix

Answer 1

The Correct Option is C

To determine the nature of \(AB - BA\), let’s use the properties of symmetric and skew-symmetric matrices.

Symmetric Matrix Property: A matrix \(M\) is symmetric if \(M^T = M\).

Since \(A\) and \(B\) are symmetric matrices, we know \(A^T = A\) and \(B^T = B\).

Now, consider \((AB - BA)^T\):

\[ (AB - BA)^T = B^T A^T - A^T B^T \]

Since \(A^T = A\) and \(B^T = B\), this becomes:

\[ (AB - BA)^T = BA - AB = -(AB - BA) \]

This result implies that \(AB - BA\) is a skew-symmetric matrix, as \((AB - BA)^T = -(AB - BA)\).

Thus, \(AB - BA\) is skew-symmetric.