Question

Let
$M=\left[\begin{array}{cc}si{n}^4\theta & -1-si{n}^2\theta \\ 1+co{s}^2\theta & co{s}^4\theta \end{array}\right]=\alpha I+\beta M^{-1}$
where $\alpha =\alpha \left(\theta \right)$ and $\beta =\beta \left(\theta \right)$ are real numbers, and $I$ is the 2 x 2 identity matrix. If
${\alpha }^{\ast }$ is the minimum of the set $\left\{\alpha \left(\theta \right):\theta \in [0,2\pi )\right\}$ and
${\beta }^{\ast }$ is the minimum of the set $\left\{\beta \left(\theta \right):\theta \in [0,2\pi )\right\}$,
then the value of ${\alpha }^{\ast }+{\beta }^{\ast }$ is

• A. $-\frac{37}{16}$
• B. $-\frac{31}{16}$
• C. $-\frac{29}{16}$
• D. $-\frac{17}{16}$

Answer 1

Answer: (C)

$M=\left[\begin{array}{cc}si{n}^4\theta & -1-si{n}^2\theta \\ 1+co{s}^2\theta & co{s}^4\theta \end{array}\right]=\alpha l+\beta M^{-1}$
$\therefore det\left(M\right)=|M|=si{n}^4\theta \cdot co{s}^4\theta +si{n}^2\theta co{s}^2\theta +2$
$=\left\{{\left(si{n}^2\theta co{s}^2\theta +\frac{1}{2}\right)}^2+\frac{7}{4}\right\}$
$\left[\begin{array}{cc}si{n}^4\theta & -1-si{n}^2\theta \\ 1+co{s}^2\theta & co{s}^4\theta \end{array}\right]=\left[\begin{array}{cc}\alpha & 0\\ 0& \alpha \end{array}\right]+\frac{\beta }{|M|}\left[\begin{array}{cc}co{s}^4\theta & 1+si{n}^2\theta \\ -1-co{s}^2\theta & si{n}^4\theta \end{array}\right]$
$\therefore \alpha =co{s}^4\theta +si{n}^4\theta =1-\frac{1}{2}\left(si{n}^{2}2\theta \right)$
and $\beta =-\left|M\right|$
$=-\left\{{\left(si{n}^2\theta \cdot co{s}^2\theta +\frac{1}{2}\right)}^2+\frac{7}{4}\right\}$
$\therefore {\alpha }_{min}=\frac{1}{2}$ and ${\beta }_{min}=-\frac{37}{16}$
$\therefore {\alpha }^{\ast }+{\beta }^{\ast }=\frac{1}{2}-\frac{37}{16}=-\frac{29}{16}$