Math Is Fun Forum

EPhillips1989

Member

Registered: 2007-11-03

Posts: 29

matrices

if  a,b are nxn matrices where AB=In
does anyone know how i can prove AB is invertible if and only if A and B are both invertible??

please help will be very much appreciated xx

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: matrices

here's how.

show AB is invertible if A and B are invertible.

Well we know A and B have an inverse so if we multiply AB on the right by (B^-1) (A^-1) we have AB(B^-1)(A^-1) = AI(A^-1) = A(A^-1) = I.

now show that AB is not invertible if A and B are not invertible.

we have A and B are not invertible, now suppose AB is invertible, then there is some matrix M such that
(AB)M = I, but this means that
A(BM) = I, which means A has an inverse 'BM' which is a contradiction. Therefore, AB cannot have an inverse, and is not invertible.

Last edited by mikau (2007-11-23 08:24:36)

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