If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant

$$\left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr {\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr {\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr } } \right|,$$ is

If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta & \alpha \cr } } \right]$$, then

If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then

$$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr } } \right|$$ is equal to

If the system of linear equations
$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
has a non - zero solution, then $$a, b, c$$.