Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,

I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) $$ \le $$ rank(A) + rank(B)
IV. det(A + B) $$ \le $$ det(A) + det(B)

Which of the above statements are TRUE?

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$$

Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$

Consider the following statements.

$$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not have an inverse
$$\left( {\rm II} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ has a repeated eigenvalue
$$\left( {\rm III} \right)$$ $$\,\,\,\,\,\,\,\,\,$$ $$P$$ cannot be diagonalized

Which one of the following options is correct?

If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} - 4{\lambda ^2} + a\lambda + 30.\,a \in R,$$ and one eigenvalue of $$M$$ is $$2,$$ then the largest among the absolute values of the eigenvalues of $$M$$ is ________.