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mikisims
07.03.2020 •
Mathematics
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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Ответ:
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.
Ответ:
4. Vertical angles theorem
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