Consider a matrix $$A = u{v^T}$$ where $$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ 1 \cr 1 \cr } } \right).$$ Note that $${v^T}$$ denotes the transpose of $$v.$$ The largest eigenvalue of $$A$$ is _____.

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 _______.

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

Consider the system, each consisting of m linear equations in $$n$$ variables.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution

Which one of the following is CORRECT?