Pretty straight forward Linear Algebra problem. It is really just one problem, the parts are more of a guide to solving the whole proof rather than separate problems themselves. Thank you!

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Answered Same DayDec 22, 2021

Answer To: Pretty straight forward Linear Algebra problem. It is really just one problem, the parts are more of...

Robert answered on Dec 22 2021

132 Votes

Question: Suppose that m≠n are positive integers, and A is an m
´
n matrix and B is n
´
m matrix such that AB = Im
Solution: To prove m If m>n then definitely AB can not be identity matrix of m
´
m .
This we can understand by taking a random example
Let A is an 3
´
2 matrix and B is 2
´
3 matrix such that
A =
23
34
52
éù
êú
êú
êú
ëû
and B =
ace
bdf
éù
êú
ëû
Let if possible A.B = I3
23
34
52
éù
êú
êú
êú
ëû
.
ace
bdf
éù
êú
ëû
=
100
010
001
éù
êú
êú
êú
ëû
232323
343434
525252
abcdef
abcdef
abcdef
+++
éù
êú
+++
êú
êú
+++
ëû
=
100
010
001
éù
êú
êú
êú
ëû
2a + 3b = 1
3a + 4b =0
5a + 2b = 0
The above linear equations has no solutions
Therefore
23
34
52
éù
êú
êú
êú
ëû
.
ace
bdf
éù
êú
ëû
=
100
010
001
éù
êú
êú
êú
ëû
is not possible for any values of a,b,c,d,e and f.
Hence m can’t be more than n
Conversely if we consider mLet A is an 2
´
3 matrix...

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Pretty straight forward Linear Algebra problem. It is really just one problem, the parts are more of a guide to solving the whole proof rather than separate problems themselves. Thank you!

Answer To: Pretty straight forward Linear Algebra problem. It is really just one problem, the parts are more of...

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