The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}$$, where I is 2 $$\times$$ 2 identity matrix, is :

If the system of linear equations

2x + y $$-$$ z = 3

x $$-$$ y $$-$$ z = $$\alpha$$

3x + 3y + $$\beta$$z = 3

has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.

Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 2⁴¹, then the value of det(A²) equal __________.

If $$A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]$$ and M = A + A² + A³ + ....... + A²⁰, then the sum of all the elements of the matrix M is equal to _____________.