If $$\left| {\matrix{ { - 1} & 7 & 0 \cr 2 & 1 & { - 3} \cr 3 & 4 & 1 \cr } } \right| = A$$, then $$\left| {\matrix{ {13} & { - 11} & 5 \cr { - 7} & { - 1} & {25} \cr { - 21} & { - 3} & { - 15} \cr } } \right|$$ is

If $${S_r} = \left| {\matrix{ {2r} & x & {n(n + 1)} \cr {6{r^2} - 1} & y & {{n^2}(2n + 3)} \cr {4{r^3} - 2nr} & z & {{n^3}(n + 1)} \cr } } \right|$$, then the value of

$$\sum\limits_{r = 1}^n {{S_r}} $$ is independent of

If the following three linear equations have a non-trivial solution, then

x + 4ay + az = 0

x + 3by + bz = 0

x + 2cy + cz = 0

The least positive integer n such that $${\left( {\matrix{ {\cos \pi /4} & {\sin \pi /4} \cr { - \sin {\pi \over 4}} & {\cos {\pi \over 4}} \cr } } \right)^n}$$ is an identity matrix of order 2 is