Question

Let A, B, C, D be real matrices such that $A^T=BCD;B^T=CDA;C^T=DAB$ and $D^T=ABC$ for the matrix $M=ABCD$, then find $M^{2016}$?

• A. $M$
• B. $M^2$
• C. $M^3$
• D. $M^4$

Answer 1

Answer: (A)

$M$

$M=ABCD$

$=A\left(BCD\right)$

$=A{A}^T$ ...... (1)

$M^{2016}=\left(ABCD\right)\left(ABCD\right)\left(ABCD\right).......\left(ABCD\right)$

$=\left(ABC\right)\left(DAB\right)\left(CDA\right)\left(BCD\right).....\left(BCD\right)$

$=D^T{C}^T{B}^T{A}^T..........A^T$

$=(BCD{)}^T{A}^T=A{A}^T$ ........ (2)

From (1) and (2),

$M=M^{2016}$