Math Problem Statement

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).

Solution

To solve for the determinant of the matrix AnA_n, where AnA_n is defined as (ai,j)(a_{i,j}) with ai,j=2δi,jδi+1,jδi,j+1a_{i,j} = 2 \delta_{i,j} - \delta_{i+1,j} - \delta_{i,j+1}, we need to interpret the given components correctly. Here, δi,j\delta_{i,j} is the Kronecker delta, which is 1 if i=ji = j and 0 otherwise.

Let's start by constructing the matrix AnA_n explicitly for a general nn. The matrix AnA_n is tridiagonal with the following structure:

  • The main diagonal entries are all 2.
  • The subdiagonal and superdiagonal entries are all -1.

For example, for n=3n = 3, the matrix A3A_3 would be:

2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$$ In general, for any $$n$$, the matrix $$A_n$$ looks like: \[ A_n = \

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Determinants
Tridiagonal Matrices
Kronecker Delta

Formulas

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Theorems

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Suitable Grade Level

Advanced Mathematics