126 questions linked to/from If $AB = I$ then $BA = I$
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### Matrices: left inverse is also right inverse? [duplicate]

If $A$ and $B$ are square matrices, and $AB=I$, then I think it is also true that $BA=I$. In fact, this Wikipedia page says that this "follows from the theory of matrices". I assume there's a nice ...
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Given a square matrix $A$ that has full row rank we know that the matrix is invertible. So there is a matrix $B$ such that $$AB=1$$ writing this in component notation, $$A_{ij}B_{jk}=\delta_{ik} ... 4answers 751 views ### Are there matrices such that AB=I and BA \neq I [duplicate] Are there matrices such that AB=I and BA \neq I ? A and B are square matrices 2answers 2k views ### What 's the short proof that for square matrices AB = I implies BA = I? [duplicate] Possible Duplicate: If AB = I then BA = I I'm trying to remember the one line proof that for square matrices AB = I implies BA = I. I think it uses only elementary matrix properties and ... 3answers 374 views ### Can we prove BA=E from AB=E? [duplicate] I was wondering if AB=E (E is identity) is enough to claim A^{-1} = B or if we also need BA=E. All my textbooks define the inverse B of A such that AB=BA=E. But I can't see why AB=E ... 1answer 2k views ### For square matrix, right or left inverse is equivalent to inverse. [duplicate] Definitions: Let A be an n\times n matrix. The n\times n matrix B(=A^{-1}) is an inverse for A if AB=BA=I. Let A be a k\times n matrix.The n\times k matrix B is a right ... 1answer 1k views ### Why does A times its inverse equal to the identity matrix? [duplicate] I was trying to come up with a proof of why: AA^{-1} = I. If we know that: A^{-1}A = I, then A(A^{-1}A) = A \implies (AA^{-1})A = A. However I don't like setting AA^{-1} = I for fear that it ... 1answer 1k views ### Proof that (AA^{-1}=I) \Rightarrow (AA^{-1} = A^{-1}A) [duplicate] I'm trying to prove a pretty simple problem - commutativity of multiplication of matrix and its inverse. But I'm not sure, if my proof is correct, because I'm not very experienced. Could you, please, ... 3answers 329 views ### How to prove that a matrix is invertible \iff invertible at right \iff invertible at left? [duplicate] Let A a n\times n invertible at left. In fact, I just want to prove that it's invertible at right (the rest is obvious). All what I can say is that there is a B s.t. BA=I. To prova AB=I, I ... 1answer 268 views ### For square matrices A, B, is AB=I sufficient that A and B are inverse of each other? [duplicate] Possible Duplicate: If AB = I then BA = I If A and B are two square matrices, and we know AB=I where I is the identity matrix. Is it sufficient that BA=I as well so that A and B ... 2answers 388 views ### Proving AB=BA=I [duplicate] I have seem to forgot this important fact, and I am trying to prove it to myself by looking at A as matrix of the elementary row operations. Where I seem to get stuck is that if A are elementary ... 1answer 446 views ### Does AA^T=I imply that A^TA=I? [duplicate] Does AA^T=I imply that A^TA=I? The wiki article defines the orthogonal group as:$$o(n,\Bbb C) = \{ A\in M_n(\Bbb C): AA^T=A^TA=I \}$$My book writes:$$o(n,\Bbb C) = \{ A\in M_n(\Bbb C): AA^T=...

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