The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 $$ \times $$ 2 matrix such that the trace of A is 3 and the trace of A³ is $$-$$18, then the value of the determinant of A is .............

Suppose

det$$\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0$$

holds for some positive integer n. Then $$\sum\limits_{k = 0}^n {{{{}^n{C_k}} \over {k + 1}}} $$ equals ..............

Let P be a matrix of order 3 $$ \times $$ 3 such that all the entries in P are from the set {$$-$$1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .

For a real number $$\alpha $$, if the system

$$\left[ {\matrix{ 1 & \alpha & {{\alpha ^2}} \cr \alpha & 1 & \alpha \cr {{\alpha ^2}} & \alpha & 1 \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr { - 1} \cr 1 \cr } } \right]$$

of linear equations, has infinitely many solutions, then 1 + $$\alpha $$ + $$\alpha $$² =