Jr.Inter !! 12 Let A and B be inver...
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Jr.Inter !! 12 Let A and B be invertiable matrices then May '03 show that (AB)-1 = B-1 A-1. Sol. A is invertible matrix then A-l exists and AA-1-A1A=I

JEE/Engineering Exams
Maths
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We are given with two invertible matrices ( mathrm{A} ) and ( mathrm{B} ), how to prove that ( (A B)^{-1}=B^{-1} A^{-1} ? ) We know that if A.B ( =I ) then it means ( mathrm{B} ) is inverse of matrix ( mathrm{A} ) where ( I ) is an identity matrix. If, we can prove that ( (A B) cdot B^{-1} A^{-1}=I ) then it means that ( B^{-1} A^{-1} ) is inverse of ( A B . ) In other words proving ( (A B) cdot B^{-1} A^{-1}=I ) ( Rightarrow(A B)^{-1}=B^{-1} A^{-1} ) Lets simplify ( A B . B^{-1} A^{-1} ) ( Rightarrow A I A^{-1}=A A^{-1}=I quad{A I=A ) and ( left.A A^{-1}=Iright} ) Therefore, from above equation, we can say that ( B^{-1} A^{-1} ) is inverse of ( A B ) ( Rightarrow(A B)^{-1}=B^{-1} A^{-1} ) which is the desired equation.
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