Question

Let $A,B,C,D$ be (not necessarily square) real matrices such that $A^T=BCD;B^T=CDA;C^T=DAB$ and $D^T=ABC$ for the matrix $S=ABCD$, then which of the following is/are true

• A. $S^3=S$
• B. $S^2=S^4$
• C. $S=S^2$
• D. none of these

Answer 1

Answer: (A), (B)

The correct options are
A $S^3=S$
B $S^2=S^4$

Given, $A^T=BCD$
$\Rightarrow A{A}^T=ABCD$

$\Rightarrow A{A}^T=S$

$\Rightarrow A{A}^T=S^T$
$\Rightarrow S=S^T$
$D^T{C}^T{B}^T{A}^T=ABC.DAB.CDA.BCD$
$(ABCD{)}^T=(ABCD\left)\right(ABCD\left)\right(ABCD)$
$S^T=S^3$

$\Rightarrow S=S^3$

$\Rightarrow S^2=S^4$
Hence, options A and B.