   Chapter 1.6, Problem 16E

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Textbook Problem
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# a. Prove part d of Theorem 1.30.b. Prove part e of Theorem 1.30.Theorem 1.30 □ Properties of Matrix Addition.Addition in M m × n ( R ) has the following properties.d. Each element of M m × n ( R ) has an additive inverse in M m × n ( R ) .e. Addition is commutative in M m × n ( R ) .

(a)

To determine

To prove: Addition in Mm×n() has the property that each element in Mm×n() has an additive inverse in Mm×n().

Explanation

Formula used:

Addition in Mm×n() is defined by

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

Proof:

Let A=[aij]m×n is any element in Mm×n().

Om×n denote the m×n matrices that has all elements zero.

Let A=[aij]m×n is element of Mm×n()

(b)

To determine

To prove: Addition is commutative in Mm×n().

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