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I'm reading Linear Algebra by Bill Jacob and am having trouble with his development of the theory behind the right inverse of a matrix. I did an internet search but didn't find anything useful. Does anyone know of a reference that might explain the concept a little more clearly?

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What are you asking exactly? An explanation of the concept of the right inverse of a matrix? What are you not understand about those? If you put some detail on where you're stuck maybe we could help more. Is he speaking of the Moore-Penrose inverse? Can you describe at least the definition he's using of "right inverse" so that we can maybe relate it to something else? For square matrices with field entries, it is well-known that left-inverses are right-inverses and vice-versa, so you need to be more explicit. –  Patrick Da Silva Aug 17 '11 at 1:22
    
In the book Jacob gives a four part theorem for the criteria for the existence of right inverses. He then uses the proof to demonstrate how to calculate them. I'm finding this somewhat confusing as he didn't lead up to the proof with any explanation. What I'm looking for is an explanation of the theory of right inverses that might be a little easier for me to understand. Thanks, Greg –  CritChamp Aug 17 '11 at 1:47
    
@Theo : I was asking precisions on what was "his" problem rather than "the" problem. Maybe he was asking for right-inverses of matrices that are not square or something like that. He spoke of right-inverses as if they were some particular entity, which made me believe he had some particular concept in mind, i.e. a non-trivial concept, maybe. –  Patrick Da Silva Aug 17 '11 at 2:00
    
@Theo : I couldn't imagine something else either... but I assumed OP knew left-inverses and right-inverses were the same, perhaps he should tell us if he knows this. And yes, I was trying to be helpful. –  Patrick Da Silva Aug 17 '11 at 2:18
    
@Patrick: I'm removing my comments. I didn't intend to criticize you in the least and no offense intended at all... It's Greg's task to induce someone to help him and it's unlikely that it's going to be me... –  t.b. Aug 17 '11 at 2:26
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1 Answer 1

To find a left inverse of a square matrix, you would perform row operations, and keep account of each operation by performing the corresponding operation to the identity matrix, and multiplying the result on the left of the given matrix. Each row operation is applied to the left and the composition of these (applied to the identity matrix) is a right inverse for your given matrix provided the reduced row echelon form is the identity.

By analogy a right inverse would correspond to a sequence of column operations. You can figure out what a column reduction is even if it doe not make algorithmic sense to do this. Later you will learn that the right inverse is the left inverse. Try for a few small (2x2) or (3x3) examples.

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Do you know the book? Your answer makes me wonder if that book defines right-inverses, left-inverses and doesn't yet assume that they are the same. –  Patrick Da Silva Aug 17 '11 at 1:28
    
If it does and doesn't even suggest it as an exercise,the author has no business writing textbooks,Patrick.Especially for undergraduates,who are confused to begin with. –  Mathemagician1234 Feb 14 '12 at 17:37
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