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 ________.

Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$ and $$10 \times 5,$$ respectively. The minimum number of scalar multiplications required to find the product $${A_1}{A_2}{A_3}{A_4}$$ using the basic matrix multiplication method is _________.

The value of the expression $${13^{99}}$$ ($$mod$$ $$17$$), in the range $$0$$ to $$16,$$ is ______________ .

Consider the recurrence relation $${a_1} = 8,\,{a_n} = 6{n^2} + 2n + {a_{n - 1}}.$$ Let $${a_{99}} = K \times {10^4}.$$ The value of $$K$$ is ____________.