in matrix multiplication if $AB=I$, do $A$ and $B$ need to be inverses of each other? Can $BA$ not equal I for the same two matrices?
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1What if your matrices aren't square? – Bob Krueger Apr 28 '15 at 15:51
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yes Matrices A and B needs to be inverse of each other for this fact to hold $AB=BA= I$ since we know that $AB \not=BA$ – user146269 Apr 28 '15 at 15:57
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Using the multiplicative property of determinants $detAB=detAdetB=detI$
$detI\neq 0 \implies detA\neq 0 $ and $detB\neq0$
As the determinants for $A$ and $B$ are non zero then they are not singular hence inverse exist.
$AB=I$ now multiply on the right by $B^{-1}$
$\begin{align} ABB^{-1}&=IB^{-1}\\ AI&=B^{-1}\\ A&=B^{-1} \end{align}$
A similar thing can be done by multiplying by $A^{-1}$ on the left to give $B=A^{-1}$
Karl
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