Question

Define symmetric and skew-symmetric matrices. If $A$ and $B$ are symmetric matrices, prove that $(AB-BA)$ is a skew-symmetric matrix.

Hint:

Remember: $(AB)^T = B^T A^T$. This property is often key in proving symmetry or skew-symmetry results.

Answer 1

Step 1: Definitions. - A matrix $M$ is symmetric if $M^T = M$. - A matrix $M$ is skew-symmetric if $M^T = -M$.

Step 2: Given. $A$ and $B$ are symmetric matrices. So, \[ A^T = A, B^T = B \]

Step 3: Consider $(AB - BA)$. Take transpose: \[ (AB - BA)^T = (AB)^T - (BA)^T \] \[ = B^T A^T - A^T B^T \] Since $A^T = A$ and $B^T = B$: \[ = BA - AB = -(AB - BA) \]

Step 4: Conclusion. \[ (AB-BA)^T = -(AB-BA) \implies AB-BA \text{ is skew-symmetric.} \]

Final Answer: \[ \boxed{AB - BA \; \text{is skew-symmetric if $A$ and $B$ are symmetric.}} \]