Question
Let \[M = \left[ \begin{array}{l}
\,\,\,{\sin ^4}\theta \,\,\,\,\,\,\,\,\,\,\, - 1 - {\sin ^2}\theta \\
1 + {\cos ^2}\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,{\cos ^4}\theta
\end{array} \right] = \alpha I + \beta {M^{ - 1}}\] Where $$\alpha = \alpha \left( \theta \right){\text{and }}\beta = \beta \left( \theta \right)$$ are real numbers, and $$I$$ is the $$2 \times 2$$ identity matrix. If $${a^*}$$ is the minimum of the set $$\left\{ {\alpha \left( \theta \right):\theta \in \left[ {0,2\pi } \right)} \right\}$$ and $${\beta ^*}$$ is the minimum of the set $$\left\{ {\beta \left( \theta \right):\theta \in \left[ {0,2\pi } \right)} \right\}.$$ Then the value of $${a^*} + {b^ * }$$ is
A.
$$ - \frac{{31}}{{16}}$$
B.
$$ - \frac{{17}}{{16}}$$
C.
$$ - \frac{{37}}{{16}}$$
D.
$$ - \frac{{29}}{{16}}$$
Answer :
$$ - \frac{{29}}{{16}}$$
Solution :
\[M = \left[ \begin{array}{l}
\,\,{\sin ^4}\theta {\rm{ }} - 1\sin \theta \\
{\rm{1 + co}}{{\rm{s}}^2}\theta \,\,\,\,\,\,\,\,\,\,{\cos ^4}\theta
\end{array} \right]\]
$$\eqalign{
& \left| M \right| = {\sin ^4}\theta {\cos ^4}\theta + 1 + {\sin ^2}\theta + {\cos ^2}\theta + {\sin ^2}\theta {\cos ^2}\theta \cr
& = 2 + {\sin ^2}\theta {\cos ^2}\theta + {\sin ^4}\theta {\cos ^4}\theta \cr} $$
\[{M^{ - 1}} = \frac{1}{{\left| M \right|}}\left[ \begin{array}{l}
\,\,\,{\cos ^4}\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 + {\sin ^2}\theta \\
- 1 - {\cos ^2}\theta \,\,\,\,\,\,\,\,\,\,\,{\sin ^4}\theta
\end{array} \right]\]
$${\text{Given}}\,{\text{that}}\,M = \alpha I + \beta {M^{ - 1}}$$
\[ \Rightarrow \,\left[ \begin{array}{l}
\,\,\,{\sin ^4}\theta \,\,\,\,\,\,\,\,\, - 1 - {\sin ^2}\theta \\
1 + {\cos ^2}\theta \,\,\,\,\,\,\,\,{\cos ^4}\theta
\end{array} \right] = \left[ \begin{array}{l}
\alpha\,\,\,\,\,0\\
0\,\,\,\,\,a
\end{array} \right] + \frac{\beta }{{\left| M \right|}}\left[ \begin{array}{l}
\,\,\,{\cos ^4}\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 + {\sin ^2}\theta \\
- 1 - {\cos ^2}\theta \,\,\,\,\,\,\,\,\,\,\,\,\,{\sin ^4}\theta
\end{array} \right]\]
$$\eqalign{
& \Rightarrow \,\frac{\beta }{{\left| M \right|}} = - 1\,{\text{and}}\,\alpha \, + \frac{\beta }{{\left| M \right|}}{\cos ^4}\theta = {\sin ^4}\theta \cr
& \Rightarrow \,\alpha = {\sin ^4}\theta + {\cos ^4}\theta \cr
& \Rightarrow \,\beta = - \left[ {2 + {{\sin }^2}\theta {{\cos }^2}\theta + {{\sin }^4}\theta {{\cos }^4}\theta } \right] \cr
& {\text{Now,}}\, \alpha \, = {\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)^2} - 2{\sin ^2}\theta {\cos ^2}\theta \cr
& = 1 - 2{\sin ^2}\theta {\cos ^2}\theta = 1 - \frac{1}{2}{\sin ^2}2\theta \cr} $$
For $$\alpha $$ to be minimum $${\sin ^2}2\theta $$ is maximum i.e., 1.
$$\eqalign{
& \therefore \,{\alpha ^ * } = 1 - \frac{1}{2} = \frac{1}{2} \cr
& {\text{Also}},\beta = - \left[ {2 + \frac{1}{4}{{\sin }^2}2\theta + \frac{1}{{16}}{{\sin }^4}2\theta } \right] \cr} $$
For $$\beta $$ to be minimum, $${\sin ^2}2\theta $$ is maximum i.e.,
$$\eqalign{
& \therefore \,\,{\beta ^ * } = - \left[ {2 + \frac{1}{4} + \frac{1}{{16}}} \right] = - \frac{{32 + 4 + 1}}{{16}} = \frac{{ - 37}}{{16}} \cr
& \therefore \,\,{\alpha ^ * } + {\beta ^ * } = \frac{1}{2} - \frac{{37}}{{16}} = \frac{{ - 29}}{{16}} \cr} $$