Matrix multiplication of a symmetric and skewsymmetric matrix

Question

Consider a matrix multiplication of the form $A^iB^{ij}C^{jk}D^k$, with repeated indices summed. Here $B^{ij}$ and $C^{jk}$ are respectively symmetric and skewsymmetric. Does this product vanish?

Answer 1

No...Take $B=\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$, $C=\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$, $A=(0,1)$, $D=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$. The product is $$ ABCD=-1. $$