Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........(E)

Let $$\omega$$ be a solution of $${x^3} - 1 = 0$$ with $${\mathop{\rm Im}\nolimits} (\omega ) > 0$$. If a = 2 with b and c satisfying (E), then the value of $${3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}$$ is equal to

Let a, b and c be three real numbers satisfying

$$[\matrix{ a & b & c \cr } ]\left[ {\matrix{ 1 & 9 & 7 \cr 8 & 2 & 7 \cr 7 & 3 & 7 \cr } } \right] = [\matrix{ 0 & 0 & 0 \cr } ]$$ ........ (E)

Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation ax² + bx + c = 0, then $$\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}} $$ is

Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]$$, where each of a, b, and c is either $$\omega$$ or $$\omega$$². Then the number of distinct matrices in the set S is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is