   Chapter 10.2, Problem 16ES ### 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.Prove that matrix multiplication is associative: If A,B, and C are any m × k, k × r, and r × n matrices, respectively, then (AB)C=A(BC). (Hint: Summation notation is helpful.)

To determine

Prove that matrix multiplication is associative: If A, B and C are any m × k, k × r, and r × n matrices, respectively, then (AB )C = A (BC ).

Explanation

Given information: A, B and C are any m × k, k × r, and r × n matrices, respectively.

Assume the entries of all matrices are real numbers.

Proof:

Given: A Is an m×k matrix, B Is a k×r matrix and C Is an r×n matrix.

To proof: (AB)C=A(BC)

PROOF:

Let A=(aij), B=(bij), C=(cij), AB=(dij), BC=(eij), (AB)C=(fij), A(BC)=(gij).

We can then proof that (AB)C=A(BC) by proving that fij=gij for all i = 1,2,…, m and for all

j = 1,2,…, n.

dij=ijth entry of (AB)C     =u=1rd iuc uj     =u=1r( v=1 k a iv b

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