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Let A_{n} the nn matrix such that a i, j =2 delta i,j - delta i+1,j - delta i,j+1 where delta t, j equals 1 if i = j and zero otherwise. Compute det(An).

The problem you've provided defines a matrix \( A_n \) with a specific structure based on the Kronecker delta function \( \delta_{i,j} \). We need to compute the determinant of this matrix \( A_n \).

### Matrix \( A_n \) Description
Given:
- \( A_n \) is an \( n \times n \) matrix.
- The elements of the matrix are defined as:
  \[
  a_{i,j} = 2\delta_{i,j} - \delta_{i+1,j} - \delta_{i,j+1}
  \]
where \( \delta_{i,j} \) is the Kronecker delta, meaning:
  \[
  \delta_{i,j} = \begin{cases}
  1 & \text{if } i = j, \\
  0 & \text{if } i \neq j.
  \end{cases}
  \]

### Explanation of the Matrix Structure
The matrix \( A_n \) can be described as a tridiagonal matrix with:
- Diagonal elements \( a_{i,i} = 2 \) for all \( i \).
- Subdiagonal elements \( a_{i,i+1} = -1 \) for \( i = 1, 2, \dots, n-1 \).
- Superdiagonal elements \( a_{i+1,i} = -1 \) for \( i = 1, 2, \dots, n-1 \).
- All other elements are zero.

In matrix form, \( A_n \) looks like this:
\[
A_n = \begin{pmatrix}
2 & -1 & 0 & \cdots & 0 \\
-1 & 2 & -1 & \cdots & 0 \\
0 & -1 & 2 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 2
\end{pmatrix}
\]

### Determinant of \( A_n \)
The determinant of this tridiagonal matrix is known and follows the recurrence relation:
\[
\det(A_n) = 2 \det(A_{n-1}) - \det(A_{n-2}),
\]
with initial conditions:
\[
\det(A_1) = 2, \quad \det(A_2) = 3.
\]

The closed-form expression for the determinant of \( A_n \) is given by:
\[
\det(A_n) = n + 1.
\]

### Final Answer
\[
\text{The determinant of } A_n \text{ is } \det(A_n) = n + 1.
\]

Would you like any further details or clarifications on this solution?

### Related Questions:
1. How is the determinant formula for tridiagonal matrices derived?
2. What are the properties of the Kronecker delta function?
3. How can the recurrence relation for determinants be solved using generating functions?
4. What is the significance of tridiagonal matrices in numerical analysis?
5. How does the determinant change if the diagonal elements of the matrix are altered?
6. Can the same determinant formula be applied if the matrix is not square?
7. How would the determinant be affected if an additional row or column was added?
8. What are the applications of such tridiagonal matrices in physical sciences?

### Tip:
When solving for determinants of matrices with special structures, look for patterns or properties (like symmetry or specific element placement) that simplify the computation, potentially leading to a recurrence relation or closed-form solution.

Learn how to compute the determinant of matrix A_n defined by the Kronecker delta function. Understand the structure of tridiagonal matrices and the closed-form solution for their determinant. Suitable for advanced undergraduate students studying matrix theory and determinants.

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