Suppose that P is a 4 × 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]T​. The rank of P is _________

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?

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 _______.