The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such that $$A=A^{-1}$$, is ___________.

Let $$A=\left[\begin{array}{lll}
1 & a & a \\
0 & 1 & b \\
0 & 0 & 1
\end{array}\right], a, b \in \mathbb{R}$$. If for some

$$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc}
1 & 48 & 2160 \\
0 & 1 & 96 \\
0 & 0 & 1
\end{array}\right]
$$ then $$n+a+b$$ is equal to ____________.

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+(\omega B)^{n}=A+B\right\}$$ is equal to ____________.

Let $$M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]$$, where $$\alpha$$ is a non-zero real number an $$N = \sum\limits_{k = 1}^{49} {{M^{2k}}} $$. If $$(I - {M^2})N = - 2I$$, then the positive integral value of $$\alpha$$ is ____________.