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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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Let A and B be n × n matrices over such that A 1 and B 1 Prove that ( A B ) 1 exists and that ( A B ) 1 = B 1 A 1 . (This result is known as the reverse order law of inverses.)

To determine

To prove: Let A and B be n×n matrices over such that A1 and B1 exists, then (AB)1 exists and (AB)1=B1A1.

Explanation

Given information:

A and B be n×n matrices over such that A1 and B1 exists.

Proof:

Let A and B be n×n matrices over such that A1 and B1 exists.

Therefore, AA1=A1A=I and BB1=B1B=I where IMn().

Now, by using associativity of matrix multiplication,

(AB)(B1A1)=A(BB1)A1

=AIA1 …… as BB1=B1B=I

=AA1

=I

Therefore, (AB)(

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