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Unformatted text preview: 3/2 Then the matrix associated to T -1 is A-1 : 1/(3/4 + 1/4) 3/2 1/2, 3/2 24.3 Example If S and T are two 1-1 linear maps show that ( S T )-1 = T-1 S -1 . Solution. Let A and B be the two matrices induced by S and T respectively. Since both S and T are 1-1 then both A x = 0 and B x = 0 have unique solutions and so they are both invertible. (Previous result) Since A and B are both invertible then A-1 is induced by S -1 and B -1 is induced by T-1 . (Previous result) Furthermore AB is invertible and so ( AB ) -1 is induced by ( S T )-1 . Since T-1 S -1 ( x ) = B -1 A-1 ( x ) = ( AB )-1 ( x ) = ( S T )-1 ( x ) for all x . Then ( S T )-1 = T-1 S -1 ....
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- Winter '08