Math Problem Statement
calculate determinent if matrix is nxn size where first row is x-1 2 2 2 ... 2 , 2 x-1 2 2 ... 2,...... 2 2 2 ..... x-1 calculate determinent
Solution
We are tasked with calculating the determinant of an matrix where:
- The diagonal elements of the matrix are .
- The off-diagonal elements are .
The matrix can be represented as:
\begin{bmatrix} x-1 & 2 & 2 & \dots & 2 \\ 2 & x-1 & 2 & \dots & 2 \\ 2 & 2 & x-1 & \dots & 2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 2 & 2 & 2 & \dots & x-1 \end{bmatrix}.$$ --- ### Step 1: Formulate the Matrix This is a symmetric matrix of the form: $$A = (x-1)I + 2J,$$ where: - $$I$$ is the identity matrix. - $$J$$ is the matrix of all ones. ### Step 2: Key Matrix Properties 1. $$J$$ is rank 1, so it has one non-zero eigenvalue equal to $$n$$ (the sum of all elements in any row) and $$n-1$$ eigenvalues equal to $$0$$. 2. The eigenvalues of $$A$$ are given by $$(x-1) + 2\lambda$$, where $$\lambda$$ are the eigenvalues of $$J$$. The eigenvalues of $$J$$ are: $$\lambda = n \text{ (multiplicity 1)}, \quad \lambda = 0 \text{ (multiplicity } n-1 \text{)}.$$ Thus, the eigenvalues of $$A$$ are: $$\begin{cases} (x-1) + 2n & \text{(multiplicity 1)}, \\ (x-1) + 2(0) = x-1 & \text{(multiplicity } n-1 \text{)}.$$ --- ### Step 3: Determinant Formula The determinant of $$A$$ is the product of its eigenvalues: \[ \text{det}(A) = \big[(x-1) + 2n\big] \cdot \big[(x-1)\big]^{n-1}. \] Simplify: $$\text{det}(A) = \big(x - 1 + 2n\big) \cdot (x-1)^{n-1}.$$ --- ### Final Answer: $$\boxed{\text{det}(A) = (x - 1 + 2n)(x-1)^{n-1}}$$ Let me know if you would like detailed steps for eigenvalue derivation or properties! 😊 --- ### 5 Related Questions: 1. What happens to the determinant as $$x \to 1$$? 2. How does the determinant behave when $$n$$ increases? 3. Can you prove this result using cofactor expansion? 4. What are the eigenvectors corresponding to these eigenvalues? 5. What if the off-diagonal elements were $$c$$ instead of $$2$$? ### Tip: Symmetry in matrices often allows simplifying determinant calculations using eigenvalues.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Eigenvalues and Eigenvectors
Formulas
Determinant formula for a symmetric matrix: det(A) = product of eigenvalues
Eigenvalues of A = (x-1 + 2n, (x-1)^(n-1))
Theorems
Rank-1 Matrix Eigenvalue Theorem
Properties of Symmetric Matrices
Determinant as Product of Eigenvalues
Suitable Grade Level
Undergraduate Mathematics