To determine whether the statement "If $A$ and $B$ are symmetric matrices, then $ABA$ is skew-symmetric" is true or false, we need to analyze the properties of symmetric and skew-symmetric matrices.

Definitions

1. Symmetric Matrix: A matrix $A$ is symmetric if $A^T = A$.
2. Skew-Symmetric Matrix: A matrix $C$ is skew-symmetric if $C^T = -C$.

Given

• $A$ and $B$ are symmetric matrices, i.e., $A^T = A$ and $B^T = B$.

To Prove

We need to prove whether $(ABA)^T = -ABA$.

Proof

Let's find the transpose of $ABA$: $(ABA)^T = A^T B^T A^T$

Since $A$ and $B$ are symmetric: $A^T = A \quad \text{and} \quad B^T = B$

Substituting these into the equation, we get: $(ABA)^T = A B A$

Therefore: $(ABA)^T = ABA$

This implies that $ABA$ is symmetric, not skew-symmetric.

Conclusion

Since $(ABA)^T = ABA$, which is not equal to $-ABA$, the matrix $ABA$ is not skew-symmetric.

Therefore, the statement is False.

The assertion that $ABA$ is skew-symmetric when $A$ and $B$ are symmetric matrices is incorrect. Instead, $ABA$ remains symmetric.

If you have further questions or need more details, feel free to ask!

Here are 8 related questions you might find interesting:

1. What is the result when a symmetric matrix is multiplied by a skew-symmetric matrix?
2. Can a matrix be both symmetric and skew-symmetric? If yes, what does that imply?
3. How do the eigenvalues of a symmetric matrix differ from those of a skew-symmetric matrix?
4. What is an example of a 2x2 skew-symmetric matrix?
5. How can you determine if a matrix is symmetric by inspecting its entries?
6. What are the properties of the transpose operation when applied to the product of matrices?
7. Can a non-square matrix be symmetric?
8. How does the determinant of a symmetric matrix relate to its eigenvalues?

Tip: When dealing with matrix transposes, remember that the transpose of a product of matrices reverses the order: $(AB)^T = B^T A^T$.