Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0} $$ where $${{a_i}}$$ are column vectors in $${R^{11}}.$$ Consider the set of linear equations $$AX=b$$

Where $$A = \left[ {{a_1},.....,\,\,{a_n}} \right]$$ and $$b = \sum\limits_{i = 1}^n {{a_i}.} $$
The set of equations has

Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \cr } } \right]$$ and $$Q = \left[ {\matrix{ { - 1} & { - 2} & { - 1} \cr 6 & {12} & 6 \cr 5 & {10} & 5 \cr } } \right]$$ be two matrices.
Then the rank of $$P+Q$$ is _______.

Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determinant of $$P$$ is _______.

Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following is the recurrence relation for $${a_n}$$?