If P is a 3 × 3 real matrix such that P ^ T = aP + ( a - 1 ) I ,...

Given,

${P}^{T} = aP + ( a - 1 ) I . . . . . . ( 1 )$

Taking transpose on both sides we get,

${\left({P}^{T}\right)}^{T} = {\left(aP + ( a - 1 ) I\right)}^{T}$

$⇒ P = a{P}^{T} + ( a - 1 ) I . . . . . . . ( 2 )$

Subtract equation (1) and (2),

$⇒ {P}^{T} - P = a\left(P - {P}^{T}\right)$

$⇒ P = {P}^{T}$ as $a \ne - 1$

From equation (1),

$P = aP + ( a - 1 ) I$

$⇒ P = - I$

Taking adjoint on both sides,

$⇒ AdjP = Adj ( - I )$

$⇒ \left|AdjP\right| = \left|Adj ( - I )\right| = {\left|- I\right|}^{2}$

$⇒ | Adj P | = 1$