Let $\beta$ be a real number. Consider the matrix

$$ A=\left(\begin{array}{ccc} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{array}\right) $$

If $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then the value of $9 \beta$ is _________.

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 ............ .