How to prove these two statements?

Question

Let $A,B,C,D$ be real matrices (not necessarily square) such that $$A^T=BCD$$$$B^T=CDA$$$$C^T=DAB$$$$D^T=ABC.$$ For the matrix $S=ABCD$, prove that

$$S^3=S$$ and $$S^2=S^4.$$

My little brother got this in a test yesterday. I have read the homework policy here, but I seriously have no idea how to get on with this one.

Answer 1

We start by showing that $S=S^T$: \begin{align}S&=ABCD=A(BCD)=AA^T\\ S^T&=(AA^T)^T=(A^T)^TA^T=AA^T\\ \therefore S&=S^T\end{align}

Now, we give an expression for $S^3$: \begin{align}S^3&=SSS\\ &=(ABCD)(ABCD)(ABCD)\\ &=(ABC)(DAB)(CDA)(BCD)=D^TC^TB^TA^T\\ &=(ABCD)^T\\ &=S^T\\ &=S\end{align}

Multiplying each side by $S$ yields the expected result, i.e.: $$S^4=S^2$$