If P is a 3 × 3 real matrix such that P T = a P + ( a - 1 ) I , where a > 1 , then

Step-by-step solution

Given, P T = a P + ( a - 1 ) I . . . . . . ( 1 ) Taking transpose on both sides we get, P T T = aP + ( a - 1 ) I T ⇒ P = a P T + ( a - 1 ) I . . . . . . . ( 2 ) Subtract equation (1) and (2), ⇒ P T - P = a P - P T ⇒ P = P T as a ≠ - 1 From equation (1), P = a P + ( a - 1 ) I ⇒ P = - I Taking adjoint on both sides, ⇒ Adj   P = Adj ( - I ) ⇒ Adj   P = Adj   ( - I ) = - I 2 ⇒ | Adj P | = 1