Let $\mathrm{M}=\left[\begin{array}{cc}{\sin }^4\mathrm{\theta }& -1-{\sin }^2\mathrm{\theta }\\ 1+{\cos }^2\mathrm{\theta }& {\cos }^4\mathrm{\theta }\end{array}\right]=\mathrm{\alpha }\mathrm{I}+\mathrm{\beta }{\mathrm{M}}^{-1},$ where $\mathrm{\alpha }=\mathrm{\alpha }\left(\mathrm{\theta }\right)$ and $\mathrm{\beta }=\mathrm{\beta }\left(\mathrm{\theta }\right)$ are real number, and $\mathrm{I}$ is the $2\times 2$ identity matrix. If ${\mathrm{\alpha }}^{\mathrm{*}}$ is the minimum of the set { $\mathrm{\alpha }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\}$ and ${\mathrm{\beta }}^{\mathrm{*}}$ is the minimum of the set $\left\{\mathrm{\beta }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\right\},$ then the value of ${\mathrm{\alpha }}^{\mathrm{*}}+{\mathrm{\beta }}^{\mathrm{*}}$ is

Method I
As given, $M=\left[\begin{array}{cc}{\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta & -1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \\ 1+{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta & {\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta \end{array}\right]=\alpha I+\beta M^{-1}$
$adj\left(M\right)=\left[\begin{array}{cc}{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta & 1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \\ -1-{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta & {\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta \end{array}\right]$
$M^{-1}=\frac{1}{\left|M\right|}\left[\begin{array}{cc}{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta & 1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \\ -1-{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta & {\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta \end{array}\right]$
$\left[\begin{array}{cc}{\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta & -1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \\ 1+{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta & {\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta \end{array}\right]=\alpha \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\frac{\beta }{\left|M\right|}\left[\begin{array}{cc}{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta & 1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \\ -1-{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta & {\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta \end{array}\right]$
Compare element ${\mathrm{a}}_{12}$ in L.H.S. and R.H.S.
$-1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta =0+\frac{\beta }{\left|M\right|}\left(1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \right)$
$\beta =\left|M\right|$ ...(iii)
Compare element $a_{11}$ in LHS. and R.H.S.
${\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta =\alpha +\frac{\beta }{\left|M\right|}\left({\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta \right)$
From equation (iii)
${\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta =\alpha -{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta$
$\alpha ={\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta +{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta$ ...(iv)
Now from equation (iv)
$\alpha ={\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta +{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta$
$\alpha ={\left({\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta +{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta \right)}^2-2{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta$
$\alpha =1-\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta }{2}$
$0\le {\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta \le 1$
$-\frac{1}{2}\le \frac{-{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta }{2}\le 0$
$\frac{1}{2}\le 1-\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta }{2}\le 1$
$\alpha \in \left[\frac{1}{2},1\right]$ , ${\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}}=\frac{1}{2}=\alpha *$
For $\beta$ , consider
$\left|M\right|={\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta +{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +2$
$\left|M\right|\left\{{\left({\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +\frac{1}{2}\right)}^2+\frac{7}{4}\right\}$
$\left|M\right|\left\{{\left(\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta }{4}+\frac{1}{2}\right)}^2+\frac{7}{4}\right\}$
$0\le {\mathrm{s}\mathrm{i}\mathrm{n}}^{2}2\theta \le 1$
Hence, $\left|M\right|\in \left[2,\frac{37}{16}\right]$
$\beta =-\left|M\right|$
$\beta \in \left[-\frac{37}{16},-2\right]$
${\beta }_{\mathrm{m}\mathrm{i}\mathrm{n}}=-\frac{37}{16}=\beta *$
$\alpha *+\beta *=\frac{1}{2}-\frac{37}{16}=-\frac{29}{16}$
Method II
$M=\alpha I+\beta M^{-1}$
Multiply with $M$ both side
$M^2=\alpha M+\beta I$
$M^2-\alpha M-BI=0$ ...(i)
Equation (i) represents characteristic equation of matrix $‘M’$
Hence, Trace $\left(M\right)=\alpha$
$\left|M\right|=-\beta$
Sum of roots of characteristic equation $=$ Trace of matrix
Product of roots of characteristic equation $=$ determined of matrix
$\therefore \alpha ={\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta +{\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta$
$\beta =-\left({\mathrm{s}\mathrm{i}\mathrm{n}}^4\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^4\theta +{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +2\right)$
Now proceed as Method 1