Math Problem Statement
If A = [[2, 0, 2], [0, 2, 0], [0, 0, 2]], then the value of A^5 is?
Solution
The given matrix is:
2 & 0 & 2 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}.$$ ### Step-by-Step Solution: 1. **Matrix properties**: Notice that $$A$$ is an upper triangular matrix with eigenvalues $$2, 2, 2$$ (the diagonal entries). 2. **Powers of a diagonal/triangular matrix**: The property of triangular matrices allows us to compute powers efficiently. Raising $$A$$ to a power $$k$$ results in: $$A^k = \begin{bmatrix} 2^k & 0 & k \cdot 2^{k-1} \cdot 2 \\ 0 & 2^k & 0 \\ 0 & 0 & 2^k \end{bmatrix}.$$ The general pattern of the non-zero entries remains the same. 3. **Compute $$A^5$$**: Using the pattern above: $$A^5 = \begin{bmatrix} 32 & 0 & 5 \cdot 16 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{bmatrix}.$$ 4. **Factor out $$A$$**: The matrix $$A^5$$ will still be proportional to $$A$$, because the structure of $$A^5$$ is the same. Hence: $$A^5 = 32A.$$ Thus, the correct answer is **(A) 32A**. Let me know if you want further clarification or details! --- Here are 5 related questions: 1. How are powers of triangular matrices generally calculated? 2. What is the relationship between eigenvalues and matrix powers? 3. Can this property be generalized to non-triangular matrices? 4. How does the result differ for non-diagonalizable matrices? 5. What are the key properties of eigenvectors in triangular matrices? **Tip**: When working with triangular matrices, always start by examining the diagonal elements, as they often reveal significant properties of the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Powers
Triangular Matrices
Formulas
For a triangular matrix A, A^k has diagonal entries (diagonal_entry)^k
Matrix multiplication properties for triangular matrices
Theorems
Power of a triangular matrix theorem: The diagonal entries raised to the power correspond to the power of the matrix
Suitable Grade Level
Undergraduate (Linear Algebra Course)