Let A and B be nxn matrices such that AB=I n . a) Let A and B be nxn matrices su

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Let A and B be nxn matrices such that AB=In. a) Let A and B be nxn matrices such that AB isinvertible. Prove that A and B are invertible Give an exampleto show that arbitrary matrices A and B need not to be invertibleif AB is invertible. b) Prove A=B-1 (and hence B=A-1). (We are, in effect, saying that for square matrices a one-sidedinverse is a two-sided inverse.) c) State and prove analogous results for lineartransformations defined on finite-dimensional vector spaces. Let A and B be nxn matrices such that AB=In. a) Let A and B be nxn matrices such that AB isinvertible. Prove that A and B are invertible Give an exampleto show that arbitrary matrices A and B need not to be invertibleif AB is invertible. b) Prove A=B-1 (and hence B=A-1). (We are, in effect, saying that for square matrices a one-sidedinverse is a two-sided inverse.) c) State and prove analogous results for lineartransformations defined on finite-dimensional vector spaces.

Explanation / Answer

QuestionDetails: Let A and B be nxn matrices such that AB=In. a) Let A and B be nxn matrices such that AB isinvertible. Prove that A and B are invertible
Give an example to show that arbitrary matrices A and B need not tobe invertible if AB is invertible. b) Prove A=B-1 (and hence B=A-1). (We are, in effect, saying that for square matrices a
one-sided inverse is a two-sided inverse.) c) State and prove analogous results for lineartransformations defined on finite-dimensional vector spaces.
Let A and B be nxn matrices such that AB=In.PERTAINS TOPART B. I THINK. IT
IS NOT IN CONFORMITY WITH PART A....ASSUMING SO.
a) Let A and B be nxn matrices such that AB is invertible. Prove that A and B are invertible
Give an example to show that arbitrary matrices A and B need not tobe invertible if AB is invertible.
LET AB=C
A MATRIX IS INVERTIBLE IF AND ONLY IF IT IS NOT SINGULAR.
.THAT IS ITS DETERMINANT IS NOT EQUAL TO ZERO.
HENCE
|C|=|AB|=|A||B| IS NOT EQUAL TO ZERO .
HENCE
|A| IS NOT EQUAL TO ZERO .HENCE A IS INVERTIBLE.....SIMILARLY
|B| IS NOT EQUAL TO ZERO .HENCE B IS INVERTIBLE.........PROVED.
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b) Prove A=B-1 (and hence B=A-1). (Weare, in effect, saying that for square matrices a
one-sided inverse is a two-sided inverse.)
AB=I ...GIVEN
RIGHT MULTIPLYING WITH B-1,WE GET
ABB-1=IB-1=B-1......
AI=B-1......
A=B-1......PROVED
SIMILARLY LEFT MULTIPLYING WITH A WE GET
A-1AB=A-1I=A-1...
IB=A-1...
B=A.......PROVED
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c) State and prove analogous results for linear transformationsdefined on finite-dimensional vector spaces.
THIS IS A STANDARD THEOREM WHICH STATES
IF T IS A LINEAR OPERATOR ON A FINITE DIMENSIONAL VECTOR SPACE
THEN T IS INVERTIBLE IF T IS NON SINGULAR OR KERNEL OF T =0.
THIS HOLDS FOR SQUARE MATRIX A WHERE A IS LINEARE OPERATOR ONRN
PLEASE POST SEPARATELY IF YOU WANT PROOF .

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