   Chapter 1.6, Problem 32E

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# Let A and B be n     ×     n matrices over ℝ 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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