Matrices And Determinants

If P is a 3 \(\times\) 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 \(\times\) 3 identity matrix, then there exists a column matrix \(X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]\) such that