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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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a. Prove part a. of Theorem 1.34.

b. Prove part b. of Theorem 1.34.

Theorem 1.34 Distributive Properties

Let A be an m × n matrix over R , let B and C be n × p matrices over R , and let D be a p × q matrix over R . Then

A ( B + C ) = A B + A C , and

( B + C ) D = B D + C D

(a)

To determine

To prove: Let A be m×n matrix in , and let B and C are n×p matrices over . Then, A(B+C)=AB+AC.

Explanation

Given information:

A be m×n matrix in , and B,C are n×p matrices over .

Formula used:

(1) Addition in Mm×n() is defined by

[aij]m×n+[bij]m×n=[cij]m×n, where cij=aij+bij.

(2) Definition of matrix multiplication:

The product of m×n matrix A over and n×p matrix B over is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

columnjofBcolumnjofCrowiofA[ai1ai2ai3ain][b1jb2jb3jbnj]=[cij]rowiofC

Where cij=ai1b1j+ai2b2j++ainbnj.

Proof:

Let A=[aik]m×n,B=[bkj]n×p and C=[ckj]n×p are matrices over

(b)

To determine

To prove: Let B and C are n×p matrices over , and let D be a p×q matrix over . Then (B+C)D=BD+CD.

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