Math Problem Statement
If A = [[1, 1, 1], [1, 1, 1], [1, 1, 1]], what will be the value of the sum of the diagonal elements of A^5?
Solution
To solve the given problem:
Matrix :
1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ We are asked to find the sum of the diagonal elements (trace) of $$A^5$$. ### Step-by-step solution: 1. **Eigenvalues of $$A$$:** - $$A$$ is a $$3 \times 3$$ matrix where all elements are 1. Its rank is 1 (all rows are linearly dependent). - The eigenvalues of $$A$$ can be computed by observing that the sum of all rows gives the eigenvalue 3, and the remaining two eigenvalues are 0. - Eigenvalues of $$A$$: $$\lambda_1 = 3, \lambda_2 = 0, \lambda_3 = 0$$. 2. **Behavior of eigenvalues under powers:** - When a matrix is raised to a power $$n$$, its eigenvalues are raised to the power $$n$$. - Eigenvalues of $$A^5$$: $$\lambda_1^5 = 3^5 = 243, \lambda_2^5 = 0^5 = 0, \lambda_3^5 = 0^5 = 0$$. 3. **Sum of diagonal elements (trace):** - The trace of a matrix is the sum of its eigenvalues. - Trace of $$A^5 = \lambda_1^5 + \lambda_2^5 + \lambda_3^5 = 243 + 0 + 0 = 243$$. ### Final Answer: The sum of the diagonal elements of $$A^5$$ is **243**. --- Do you want a detailed explanation of any step or concept? ### Related Questions: 1. How do you compute eigenvalues for symmetric matrices? 2. Why are the eigenvalues $$3, 0, 0$$ for matrix $$A$$? 3. How does the trace of a matrix relate to its eigenvalues? 4. Can you calculate $$A^n$$ directly without eigenvalues for small powers of $$n$$? 5. What is the significance of rank in determining eigenvalues? ### Tip: For matrices with all identical rows or columns, eigenvalue decomposition simplifies calculations greatly, as the matrix is effectively rank 1.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Eigenvalues and Eigenvectors
Matrix Powers
Trace of a Matrix
Formulas
Trace(A) = sum of diagonal elements of A
For matrix A with eigenvalues λ1, λ2, ..., λn, eigenvalues of A^k are λ1^k, λ2^k, ..., λn^k
Theorems
Trace of a matrix is equal to the sum of its eigenvalues
Eigenvalue behavior under matrix powers: λ^n for eigenvalue λ
Suitable Grade Level
Grades 11-12 or Undergraduate Level