   Chapter 10.2, Problem 18ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 14—18, assume the entries of all matrices are real numbers. 18. Use mathematical induction to prove that if A is an m × m symmetric matrix, then for any integer n ≥ 1 ,   A n is also symmetric.

To determine

To prove: If A Is an m×m matrix symmetric matrix, then for any integer n1. An Is also symmetric.

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: An is a symmetric matrix for all integers n1

PROOF:

Let P(n) be "An is a symmetric matrix"

Basis step n = 1

An = A1 = AThus P(1) is true, since we have been given that A is a symmetric matrix.

Inductive step:

Let P(k) be true, thus Ak=(bij) is a symmetric matrix.

We need to prove that P ( k + 1 ) is true. Let Ak+1=(cij).

Since A = (a ij) is an m×m symmetric matrix:                               aij=aji for all i,j=1,2,...,mSince Ak = (b ij) is an m×m symmetric matrix:                              bij=bji for all i,j=1,2,...,mLet us determine if the elements of Ak+1=(c ij) also satisfy the symmetric property (c ij=c ji)

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