Let A and B be square matrices. Show that if ABequals=I then A and B are both invertible, with Bequals=Upper A Superscript negative 1A-1 and Aequals=Upper B Superscript negative 1B-1.

Question

Mathematics

Milo Francis

5 February, 23:36

Let A and B be square matrices. Show that if ABequals=I then A and B are both invertible, with Bequals=Upper A Superscript negative 1A-1 and Aequals=Upper B Superscript negative 1B-1.

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Answers (1)

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Alonzo Leblanc · Answer 1

Step-by-step explanation:

Since AB=I, we have

det (A) det (B) = det (AB) = det (I) = 1.

This implies that the determinants det (A) and det (B) are not zero.

Hence A, B are invertible matrices: A-1, B-1 exist.

Now we compute

I=BB-1=BIB-1=B (AB) B-1=BAI=BA. since AB=I

Hence we obtain BA=I.

Since AB=I and BA=I, we conclude that B=A-1.