Math Problem Statement
Please solve exercise 3
Please solve exercise 3
Solution
Let's analyze and solve each determinant problem step-by-step.
Problem 1: Determinant with Repeated Rows
The determinant is:
\begin{vmatrix} a_1 & a_2 & \cdots & a_n \\ a_1 & a_1 & \cdots & a_1 \\ \vdots & \vdots & \ddots & \vdots \\ a_1 & a_1 & \cdots & a_1 \end{vmatrix}.$$ **Observation**: All rows except the first are identical. If any two rows in a determinant are identical, the determinant is 0. **Answer**: $$\text{det} = 0$$ --- ### Problem 2: Determinant with Diagonal Elements $$a+b$$ The determinant is: $$\text{det} = \begin{vmatrix} a+b & a & \cdots & a \\ a & a+b & \cdots & a \\ \vdots & \vdots & \ddots & \vdots \\ a & a & \cdots & a+b \end{vmatrix}.$$ This matrix is symmetric and has the form: $$A = (a+b)I + aJ,$$ where: - $$I$$ is the identity matrix, - $$J$$ is the matrix of all ones. **Eigenvalues**: The eigenvalues of $$A$$ can be determined from its structure: 1. One eigenvalue is $$(a+b) + na$$, corresponding to the vector of all ones. 2. The remaining $$n-1$$ eigenvalues are $$a+b$$, corresponding to vectors orthogonal to the vector of all ones. **Determinant**: The determinant is the product of all eigenvalues: $$\text{det} = [(a+b) + na] \cdot (a+b)^{n-1}.$$ **Answer**: $$\text{det} = [(a+b) + na](a+b)^{n-1}.$$ --- ### Problem 3: Determinant with $$\lvert i-j \rvert$$ The determinant is defined as: $$A_{ij} = \lvert i-j \rvert, \quad 1 \leq i,j \leq n.$$ **Observation**: The matrix is symmetric and Toeplitz, meaning the entries depend only on $$|i-j|$$. Explicitly, the structure is: $$A = \begin{bmatrix} 0 & 1 & 2 & \cdots & n-1 \\ 1 & 0 & 1 & \cdots & n-2 \\ 2 & 1 & 0 & \cdots & n-3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n-1 & n-2 & n-3 & \cdots & 0 \end{bmatrix}.$$ For this matrix, it is known (via advanced properties of Toeplitz matrices) that the determinant is: $$\text{det}(A) = (-1)^{n-1}(n-1).$$ **Answer**: $$\text{det} = (-1)^{n-1}(n-1).$$ --- ### Final Answers: 1. $$\text{det} = 0$$ 2. $$\text{det} = [(a+b) + na](a+b)^{n-1}$$ 3. $$\text{det} = (-1)^{n-1}(n-1)$$ Would you like further clarification or detailed steps for any of these? Here are some related questions to deepen your understanding: --- 1. How can we generalize determinants with repeated rows or columns? 2. What are the eigenvalues of symmetric matrices in general? 3. What are Toeplitz matrices, and how are their determinants calculated? 4. How does the presence of $$a+b$$ in the diagonal modify the determinant structure? 5. Can you find the determinant of specific sizes, such as $$n=3$$, for these problems? **Tip**: For symmetric matrices, eigenvalue properties often simplify determinant calculations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Linear Algebra
Matrix Theory
Eigenvalues
Formulas
Determinant of a matrix
Eigenvalue-product formula for determinant
Properties of Toeplitz matrices
Theorems
Property of determinants with identical rows or columns (det = 0)
Eigenvalues of symmetric matrices
Structure of Toeplitz matrices
Suitable Grade Level
University Level