Explanation Given information: A Is an m × m matrix symmetric matrix. Assume the entries of all matrices are real numbers. Proof: Given: A Is an m × m symmetric matrix To proof: A n is a symmetric matrix for all integers n ≥ 1 PROOF: Let P ( n ) be "A n is a symmetric matrix " Basis step n = 1 A n = A 1 = A Thus P ( 1 ) is true, since we have been given that A is a symmetric matrix . Inductive step: Let P ( k ) be true, thus A k = ( b i j ) is a symmetric matrix . We need to prove that P ( k + 1 ) is true. Let A k + 1 = ( c i j ) . Since A = ( a i j ) is an m × m symmetric matrix: a i j = a j i for all i , j = 1 , 2 , ... , m Since A k = ( b i j ) is an m × m symmetric matrix: b i j = b j i for all i , j = 1 , 2 , ... , m Let us determine if the elements of A k + 1 = ( c i j ) also satisfy the symmetric property ( c i j = c j i )

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ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 10.2, Problem 18ES

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To prove: If A Is an m×m matrix symmetric matrix, then for any integer n≥1. Aⁿ Is also symmetric.

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Chapter 10.2, Problem 17ES

Chapter 10.2, Problem 19ES

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To prove: If A Is an m × m matrix symmetric matrix , then for any integer n ≥ 1 . A n Is also symmetric.

Solution Summary: The author explains that if A is an mtimes-m matrix symmetric matrix, then for any integer nge 1.

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To prove: If A Is an m×m matrix symmetric matrix, then for any integer n≥1. Aⁿ Is also symmetric.

Want to see the full answer?

Chapter 10 Solutions

To prove: If A Is an m × m matrix symmetric matrix , then for any integer n ≥ 1 . A n Is also symmetric.

Solution Summary: The author explains that if A is an mtimes-m matrix symmetric matrix, then for any integer nge 1.

Videos

To prove: If A Is an m×m matrix symmetric matrix, then for any integer n≥1. An Is also symmetric.

Want to see the full answer?

Chapter 10 Solutions

To prove: If A Is an m×m matrix symmetric matrix, then for any integer n≥1. Aⁿ Is also symmetric.