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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Label each of the following statements as either true or false.

The invertible elements of M n ( R ) form an abelian group with respect to matrix multiplication.

To determine

Whether the statement, “The invertible elements of Mn() form an abelian group with respect to matrix multiplication.” is true or false.

Explanation

Explanation:

Let G be the set of all invertible matrices of Mn().

Let A=[aij]n×n,B=[bij]n×nG that is A1 and B1 exist.

By the reverse order property, (AB)1=B1A1.

As multiplication of invertible matrices is invertible and A1 and B1 exist, so (AB)1 also exists.

So, ABG.

Hence, G is closed under multiplication.

Matrix multiplication is associative, so multiplication is associative in G.

As AIn=InA=A, for all AG, In is an identity element in G.

G is a set of invertible matrices in Mn(), so every matrix in G has an inverse in G.

Hence, G is a group with respect to multiplication

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