If A = [[ pmatrix 1 & & 1 - & 1 & - 1 & - & 1 pmatrix ]]; then for all [ ] [ ] [( 3 4,5 4 …

Question

If   A = $$\left[ {\begin{pmatrix} 1 & {\sin \theta } & 1  \
   { - \sin \theta } & 1 & {\sin \theta }  \
   { - 1} & { - \sin \theta } & 1  \ \end{pmatrix} } \right]$$;

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

Accepted Answer

$$\left| A \right| = \left| {\begin{pmatrix} 1 & {\sin \theta } & 1  \
   { - \sin \theta } & 1 & {\sin \theta }  \
   { - 1} & { - \sin \theta } & 1  \

\end{pmatrix} } \right|$$

= 2(1 + sin2$$\theta $$)

$$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},{{5\pi } \over 4}} \right) \Rightarrow {1 \over {\sqrt 2 }} < \sin \theta  < {1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$  0 $$ \le $$ sin2$$\theta $$ < $${1 \over 2}$$

$$ \therefore $$  $$\left| A \right| \in \left[ {2,3} \right)$$

Solutions · 1