If the matrix M_r is given by $${M_r} = \left( {\matrix{ r & {r - 1} \cr {r - 1} & r \cr } } \right)$$ for r = 1, 2, 3, ... then det (M₁) + det (M₂) + ... + det (M₂₀₀₈) =

Let $$\alpha,\beta$$ be the roots of the equation $$a{x^2} + bx + c = 0,a,b,c$$ real and $${s_n} = {\alpha ^n} + {\beta ^n}$$ and $$\left| {\matrix{ 3 & {1 + {s_1}} & {1 + {s_2}} \cr {1 + {s_1}} & {1 + {s_2}} & {1 + {s_3}} \cr {1 + {s_2}} & {1 + {s_3}} & {1 + {s_4}} \cr } } \right| = k{{{{(a + b + c)}^2}} \over {{a^4}}}$$ then $$k = $$

Let $$A = \left( {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right),B = \left( {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right)$$ and $$P\left( {\matrix{ 0 & 1 & 0 \cr x & 0 & 0 \cr 0 & 0 & y \cr } } \right)$$ be an orthogonal matrix such that $$B = PA{P^{ - 1}}$$ holds. Then

Under which of the following condition(s) does(do) the system of equations $$\left( {\matrix{ 1 & 2 & 4 \cr 2 & 1 & 2 \cr 1 & 2 & {(a - 4)} \cr } } \right)\left( {\matrix{ x \cr y \cr z \cr } } \right) = \left( {\matrix{ 6 \cr 4 \cr a \cr } } \right)$$ possesses(possess) unique solution ?