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Problem 1. a2x² + ... +a,x". Then (a) (b) (c) If A is diagonalizable, that is A = Prove that f(A) = Q (f(m₂) 0 = ODO¹. Assume a polynomial f(x) = a₁ + a₁x + 0 0 f(2₂) ⠀ 0 0 0 0 0 ƒ(2₂)) Q¹, where A eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then prove that f(ai) = Abi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A²-3A+1= -12 11

Problem 1. a2x² + ... +a,x". Then (a) (b) (c) If A is diagonalizable, that is A = Prove that f(A) = Q (f(m₂) 0 = ODO¹. Assume a polynomial f(x) = a₁ + a₁x + 0 0 f(2₂) ⠀ 0 0 0 0 0 ƒ(2₂)) Q¹, where A eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then prove that f(ai) = Abi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A²-3A+1= -12 11

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 1. 92x² + (a) (b) If A is diagonalizable, that is A = ODO¹. Assume a polynomial f(x) = ao + a₁x + +ax". Then Prove that f(A) = Q (f(n) 0 0 0 0 f(2₂) 0 0 f(2₂), where λ eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then prove that f(ai) = Abi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A² - 3A+I = -(-2₂29) -12 11
Transcribed Image Text:Problem 1. 92x² + (a) (b) If A is diagonalizable, that is A = ODO¹. Assume a polynomial f(x) = ao + a₁x + +ax". Then Prove that f(A) = Q (f(n) 0 0 0 0 f(2₂) 0 0 f(2₂), where λ eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then prove that f(ai) = Abi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A² - 3A+I = -(-2₂29) -12 11
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