Let $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$ and I be the identity matrix of order 3. If $$Q = [{q_{ij}}]$$ is a matrix such that $${P^{50}} - Q = I$$ and $${{{q_{31}} + {q_{32}}} \over {{q_{21}}}}$$ equals

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

Let $$P = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [{b_{ij}}]$$, where $${b_{ij}} = {2^{i + j}}{a_{ij}}$$ for $$1 \le i,j \le 3$$. If the determinant of P is 2, then the determinant of the matrix Q is