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The image displays a determinant \( D_n \) of a matrix. This matrix is of order \( n \) and has a specific pattern. Let’s analyze and derive a method to calculate this determinant.

Given the matrix \( D_n \):

\[
D_n = \begin{vmatrix}
\lambda & x & x & \cdots & x \\
y & \alpha & \beta & \cdots & \beta \\
y & \beta & \alpha & \cdots & \beta \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
y & \beta & \beta & \cdots & \alpha \\
\end{vmatrix}
\]

Here's the approach to solve for the determinant of this matrix:

1. **Expand the determinant using cofactor expansion along the first row**:
   \[
   D_n = \lambda \cdot \text{det}(M_{1,1}) - x \cdot \text{det}(M_{1,2}) - x \cdot \text{det}(M_{1,3}) - \cdots - x \cdot \text{det}(M_{1,n})
   \]
   where \( M_{1,j} \) is the submatrix obtained by removing the first row and \( j \)-th column.

2. **Simplify the submatrices**:
   Each of these submatrices will have the form:
   \[
   M_{1,j} = \begin{vmatrix}
   \alpha & \beta & \beta & \cdots & \beta \\
   \beta & \alpha & \beta & \cdots & \beta \\
   \vdots & \vdots & \vdots & \ddots & \vdots \\
   \beta & \beta & \beta & \cdots & \alpha \\
   \end{vmatrix}
   \]
   This is a symmetric matrix with a pattern.

3. **Recognize the recursive structure**:
   Notice that if you remove the first row and any column, the remaining submatrix has the same structure but of order \( n-1 \).

4. **Leverage properties of special matrices**:
   By observing the patterns, we might derive a recursion formula or a closed-form for the determinant.

To calculate the determinant, typically specific properties of matrices such as block matrix properties or symmetric matrix properties can be used. For such special matrices, often there is a known result that can simplify this calculation.

Would you like a detailed step-by-step solution of this determinant, or do you have specific questions about the properties or methods for finding determinants of such matrices?

Here are some additional questions you might consider:
1. Can you show how to expand this determinant using cofactor expansion?
2. What properties of symmetric matrices can simplify the calculation of this determinant?
3. Are there any special cases for specific values of \( n \) that simplify the calculation?
4. How can we derive a recursion relation for the determinant?
5. Are there known closed-form expressions for determinants of matrices with similar patterns?

Tip: When dealing with determinants of large matrices, identifying symmetries and patterns can significantly simplify the problem.

Learn how to solve determinants of symmetric matrices with a step-by-step guide. This problem involves matrix determinants, cofactor expansion, symmetric matrices, recursive relations, and closed-form expressions. Suitable for undergraduate level.

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