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 $S=ABCD$, consider the two statements.
I $S^3=S$
II $S^2=S^4$

• A. II is true but not I
• B. I is true but not II
• C. Both I and II are true
• D. Both I and II are false.

Answer 1

Answer: (B)

I is true but not II

$A^T=BCD$
$B^T=CDA$
$C^T=DAB$
$D^T=ABC$
$S=ABCD$
$S^T={\left[ABCD\right]}^T$
$=D^T{C}^T{B}^T{A}^T$
$=\left(ABC\right)\left(DAB\right)\left(CDA\right)\left(BCD\right)$
$=A\left(BCD\right)A\left(BCD\right)A\left(BCD\right)$
$=\left(A{A}^T\right)\left(A{A}^T\right)\left(A{A}^T\right)$
$={\left(A{A}^T\right)}^3$
$={\left(ABCD\right)}^3$
$=S^3$
$\therefore S^3=S^T\Rightarrow \therefore$ I is true
$S^2={\left(ABCD\right)}^2$
$=(A{A}^T{)}^2$
$S^4={\left(ABCD\right)}^4$
$={\left(A{A}^T\right)}^4$
$S^2\ne S^4\Rightarrow \therefore$ II is not true
$\therefore$ I is true but not II