If $P$ is a 3 x 3 real matrix such that $P^{T} = a P + (a - 1) I$ , where $a > 1$ ,then

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P is a singular matrix

$| AdjP | > 1$

$| AdjP | = \frac{1}{2}$

$| AdjP | = 1$

answer is D.

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Detailed Solution

$\begin{array}{l} P^{T} = a p + (a - 1) I \Rightarrow P = a P^{T} + (a - 1) I \\ \Rightarrow P = a (a P + (a - 1) I) + (a - 1) I \Rightarrow (1 - a^{2}) P = (a^{2} - 1) I \\ \Rightarrow P = - I (∵ a > 1) \Rightarrow∣ adj P ∣= 1 \end{array}$

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