Question

Let P,Q,R,S be (not necessarily square) real matrices such that
$P^T=\mathrm{QRS};Q^T=\mathrm{RSP};R^T=\mathrm{SPQ}\mathrm{and}{S}^T=\mathrm{PQR}.$
If the matrix A = PQRS, then $A^3=?$

• A. A
• B. $A^2$
• C. $A^6$
• D. $A^9$

Answer 1

The correct option is A A

Matrix A = PQRS
A = PQRS = P(QRS) = ${\mathrm{PP}}^T-\left(1\right)$
$A^3=\left(\mathrm{PQRS}\right)\left(\mathrm{PQRS}\right)\left(\mathrm{PQRS}\right)=\left(\mathrm{PQR}\right)\left(\mathrm{SPQ}\right)\left(\mathrm{RSP}\right)\left(\mathrm{QRS}\right)\mathrm{According}\mathrm{to}\mathrm{question}-\mathrm{PQR}=S^T\mathrm{SPQ}=R^T\mathrm{RSP}=Q^T\mathrm{QRS}=P^T\mathrm{now},A^3=S^T{R}^T{Q}^T{P}^T={\left(\mathrm{QRS}\right)}^T{P}^T={\left(P^T\right)}^T{P}^T\because \left(\mathrm{QRS}=P^T\right)-\left(2\right)\mathrm{from}\left(1\right)\mathrm{and}\left(2\right)A^3=A\mathrm{option}\left(a\right)\mathrm{is}\mathrm{correct}.$