`M=[{:(sin^(4)theta,-1-sin^(2)theta),(1+cos^(2)theta,cos^(4)theta):}]=alphaI+betaM^(-1)`
Where `alpha=alpha(theta)` and `beta=beta(theta)` ar real numbers and I is an identity matric of `2xx2`
if `alpha^(**)=` min of set `{alpha(theta):thetain[0.2pi)}`
and `beta^(**)=` min of set `{beta(theta):thetain[0.2pi)}`
Then value of `alpha^(**)+beta^(**)` is

Options

A. `(-37)/(16)`
B. `(-17)/(16)`
C. `(-31)/(16)`
D. `(-29)/(16)`

Explanation

To solve the given problem step by step, we will analyze the matrix \( M \) and derive the values of \( \alpha^{**} \) and \( \beta^{**} \).

### Step 1: Define the Matrix \( M \)
The matrix \( M \) is given as:
\[ M = \begin{pmatrix} \sin^4 \theta & -1 - \sin^2 \theta \\ 1 + \cos^2 \theta & \cos^4 \theta \end{pmatrix} \]

### Step 2: Find the Inverse of Matrix \( M \)
The inverse of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
First, we need to calculate the determinant \( |M| = ad - bc \).

#### Calculation of Determinant:
\[ |M| = \sin^4 \theta \cdot \cos^4 \theta - (-1 - \sin^2 \theta)(1 + \cos^2 \theta) \]
Expanding this gives:
\[ |M| = \sin^4 \theta \cos^4 \theta + (1 + \sin^2 \theta + \cos^2 \theta + \sin^2 \theta \cos^2 \theta) \]
Using \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ |M| = \sin^4 \theta \cos^4 \theta + 2 + \sin^2 \theta \cos^2 \theta \]

### Step 3: Substitute \( M \) and \( M^{-1} \) into the Equation
Given:
\[ M = \alpha I + \beta M^{-1} \]
where \( I \) is the identity matrix:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
Substituting \( M \) and \( M^{-1} \) into the equation gives:
\[ \begin{pmatrix} \sin^4 \theta & -1 - \sin^2 \theta \\ 1 + \cos^2 \theta & \cos^4 \theta \end{pmatrix} = \alpha \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \beta \cdot \frac{1}{|M|} \begin{pmatrix} \cos^4 \theta & 1 + \sin^2 \theta \\ -1 - \cos^2 \theta & \sin^4 \theta \end{pmatrix} \]

### Step 4: Equate the Elements
From the matrix equation, we can equate the corresponding elements to find \( \alpha \) and \( \beta \).

1. For the first element:
\[ \sin^4 \theta = \alpha + \frac{\beta \cos^4 \theta}{|M|} \]

2. For the second element:
\[ -1 - \sin^2 \theta = \frac{\beta (1 + \sin^2 \theta)}{|M|} \]

3. For the third element:
\[ 1 + \cos^2 \theta = -\frac{\beta (1 + \cos^2 \theta)}{|M|} \]

4. For the fourth element:
\[ \cos^4 \theta = \alpha + \frac{\beta \sin^4 \theta}{|M|} \]

### Step 5: Solve for \( \alpha \) and \( \beta \)
From the equations, we can isolate \( \alpha \) and \( \beta \).

After some algebraic manipulation, we find:
\[ \alpha = 1 - \frac{1}{2}\sin^2(2\theta) \]
\[ \beta = -|M| \]

### Step 6: Find the Minimum Values
To find \( \alpha^{**} \) and \( \beta^{**} \):
1. The minimum value of \( \alpha \) occurs when \( \sin^2(2\theta) \) is maximum, which is 1. Thus:
\[ \alpha^{**} = 1 - \frac{1}{2} = \frac{1}{2} \]

2. For \( \beta \), we need to calculate \( |M| \) at its maximum value. After evaluating, we find:
\[ \beta^{**} = -\frac{37}{16} \]

### Step 7: Calculate \( \alpha^{**} + \beta^{**} \)
Finally, we calculate:
\[ \alpha^{**} + \beta^{**} = \frac{1}{2} - \frac{37}{16} = \frac{8}{16} - \frac{37}{16} = -\frac{29}{16} \]

### Final Answer:
\[ \alpha^{**} + \beta^{**} = -\frac{29}{16} \]