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    i have learnt what the rule is (purely from getting an exam style question wrong), where I must reverse the order of inverse matrices products ... se

    http://portal.acm.org/citation.cfm?id=321353

    but why exactly? there is a difference between knowing then law, and wanting to know the proof or reason behind it
    thanks
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    In qualitative terms, the reason behind it is that the transformation AB represents the transformation B followed by the transformation A. The inverse is the transformation applied to AB which 'undoes' the transformation. In order to undo it, you must undo the transformation which you applied last, followed by the transformation which you applied first. So you multiply it by A^-1 followed by B^-1.

    Think of it like putting socks and shoes on. You put your socks on first, followed by yours shoes. In order to remove them, you must remove them in the opposite order, so you take your shoes off then you take your socks off.
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    To prove it:

    Let
     M=AB

\Rightarrow A^{-1}M = A^{-1}AB

\Rightarrow B^{-1}A^{-1}M = B^{-1}B

\Rightarrow B^{-1}A^{-1}M = I

    So B^{-1}A^{-1} is the inverse of matrix M, so it is also the inverse of AB.
 
 
 
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