If   A = $$\left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \theta } \cr { - 1} & { - \sin \theta } & 1 \cr } } \right]$$;

then for all $$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},{{5\pi } \over 4}} \right)$$, det (A) lies in the interval :

The set of all values of $$\lambda $$ for which the system of linear equations
x – 2y – 2z = $$\lambda $$x
x + 2y + z = $$\lambda $$y
– x – y = $$\lambda $$z
has a non-trivial solutions :

An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations
(1 + $$\alpha $$) x + $$\beta $$y + z = 2
$$\alpha $$x + (1 + $$\beta $$)y + z = 3
$$\alpha $$x + $$\beta $$y + 2z = 2
has a unique solution, is :

Let P = $$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right]$$ and Q = [q_ij] be two 3 $$ \times $$ 3 matrices such that Q – P⁵ = I₃.

Then $${{{q_{21}} + {q_{31}}} \over {{q_{32}}}}$$ is equal to :