Determinant and Transpose

Let the given matrix expression be represented as $$C = A B + B A$$. Taking the transpose on both sides and applying the reversal law for matrix multiplication yields:
$$C^T = (A B + B A)^T = (A B)^T + (B A)^T = B^T A^T + A^T B^T$$

Substituting the given properties $$A^T = A$$ and $$B^T = -B$$ into the expression simplifies it to:
$$C^T = (-B)(A) + (A)(-B) = -B A - A B = -(A B + B A) = -C$$

This confirms that $$C$$ is a skew-symmetric matrix. Taking the determinant of both sides, and noting that the matrix is of an odd order $$(3 \times 3)$$, gives:
$$\det(C^T) = \det(-C) \implies \det(C) = (-1)^3 \det(C) \implies \det(C) = -\det(C)$$

Rearranging this algebraic relation results in $$2 \det(C) = 0$$, which means $$\det(C) = 0$$.