Let M = [ cases ^ 4 θ & - 1 - ^ 2 θ 1 + ^ 2 θ & ^ 4 θ cases ] = + β...

Method I
As given, M = \left[\begin{cases}\sin^{4}\theta & - 1 - \sin^{2}\theta \\ 1 + \cos^{2}\theta & \cos^{4}\theta\end{cases}\right] = \alphaI + \beta{M}^{- 1}
adj\left(M\right) = \left[\begin{cases}\cos^{4}\theta & 1 + \sin^{2}\theta \\ - 1 - \cos^{2}\theta & \sin^{4}\theta\end{cases}\right]
{M}^{- 1} = \dfrac{1}{| M |}\left[\begin{cases}\cos^{4}\theta & 1 + \sin^{2}\theta \\ - 1 - \cos^{2}\theta & \sin^{4}\theta\end{cases}\right]
\left[$\begin{cases}\sin^{4}\theta & - 1 - \sin^{2}\theta \\ 1 + \cos^{2}\theta & \cos^{4}\theta\end{cases}$\right] = \alpha\left[$\begin{cases}1 & 0 \\ 0 & 1\end{cases}$\right] + \dfrac{\beta}{\left|M\right|}\left[$\begin{cases}\cos^{4}\theta & 1 + \sin^{2}\theta \\ - 1 - \cos^{4}\theta & \sin^{4}\theta\end{cases}$\right]$<br />Compare element${a}_{12}$in L.H.S. and R.H.S.<br />$- 1 - \sin^{2}\theta = 0 + \dfrac{\beta}{\left|M\right|}\left(1 + \sin^{2}\theta\right)$<br />$\beta = | M |$...(iii)<br />Compare element${a}_{11}$in LHS. and R.H.S.<br />$\sin^{4}\theta = \alpha + \dfrac{\beta}{\left|M\right|}\left(\cos^{4}\theta\right)$<br />From equation (iii)<br />$\sin^{4}\theta = \alpha - \cos^{4}\theta$<br />$\alpha = \sin^{4}\theta + \cos^{4}\theta$...(iv)<br />Now from equation (iv)<br />$\alpha = \sin^{4}\theta + \cos^{4}\theta$<br />$\alpha = {\left(\sin^{2}\theta + \cos^{2}\theta\right)}^{2} - 2\sin^{2}\theta\cos^{2}\theta$<br />$\alpha = 1 - \dfrac{\sin^{2}2\theta}{2}$<br />$0 \le \sin^{2}2\theta \le 1$<br />$- \dfrac{1}{2} \le \dfrac{- \sin^{2}2\theta}{2} \le 0$<br />$\dfrac{1}{2} \le 1 - \dfrac{\sin^{2}2\theta}{2} \le 1$<br />$\alpha \in $\left[\dfrac{1}{2}, 1\right]$ , {\alpha}_\min = \dfrac{1}{2} = \alpha *
For $\beta$ , consider
\left|M\right| = \sin^{4}\theta\cos^{4}\theta + \sin^{2}\theta\cos^{2}\theta + 2$<br />\left|M\right|\left\{{\left(\sin^{2}\theta\cos^{2}\theta + \dfrac{1}{2}\right)}^{2} + \dfrac{7}{4}\right\}$
\left|M\right|\left\{{\left(\dfrac{\sin^{2}2\theta}{4} + \dfrac{1}{2}\right)}^{2} + \dfrac{7}{4}\right\}$<br />$0 \le \sin^{2}2\theta \le 1$<br />Hence, \left|M\right| \in \left[2, \dfrac{37}{16}\right]$
$\beta = - \left|M\right|$
$\beta \in \left[- \dfrac{37}{16}, - 2\right]$
{\beta}_\min = - \dfrac{37}{16} = \beta *
$\alpha* + \beta* = \dfrac{1}{2} - \dfrac{37}{16} = - \dfrac{29}{16}$
Method II
$M = \alphaI + \beta{M}^{- 1}$
Multiply with $M$ both side
${M}^{2} = \alphaM + \betaI$
${M}^{2} - \alphaM - BI = 0$ ...(i)
Equation (i) represents characteristic equation of matrix $‘M’$
Hence, Trace $( M ) = \alpha$
\left|M\right| = - \beta$<br />Sum of roots of characteristic equation = Trace of matrix<br />Product of roots of characteristic equation = determined of matrix<br />$∴ \alpha = \sin^{4}\theta + \cos^{4}\theta$<br />$\beta = - \left(\sin^{4}\theta\cos^{4}\theta + \sin^{2}\theta\cos^{2}\theta + 2\right)
Now proceed as Method 1