For any 3 $$\times$$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $$\times$$ 3 identity matrix. Let E and F be two 3 $$\times$$ 3 matrices such that (I $$-$$ EF) is invertible. If G = (I $$-$$ EF)^$$-$$1, then which of the following statements is (are) TRUE?

Let M be a 3 $$ \times $$ 3 invertible matrix with real entries and let I denote the 3 $$ \times $$ 3 identity matrix. If M^$$-$$1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?

Let x $$ \in $$ R and let $$P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$, $$Q = \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 6 \cr } } \right]$$ and R = PQP^$$-$$1, which of the following options is/are correct?

$${P_1} = I = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_2} = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right],\,{P_3} = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_4} = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right],\,{P_5} = \left[ {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 \cr } } \right],\,{P_6} = \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \cr } } \right]$$ and $$X = \sum\limits_{k = 1}^6 {{P_k}} \left[ {\matrix{ 2 & 1 & 3 \cr 1 & 0 & 2 \cr 3 & 2 & 1 \cr } } \right]P_k^T$$

where $$P_k^T$$ denotes the transpose of the matrix P_k. Then which of the following option is/are correct?