Friday, October 13, 2006

Inverse of matrix

The inverse of a product AB of matrices A and B can be expressed in terms of A^(-1) and B^(-1). Let

C=AB.
(7)

Then

B==A^(-1)AB==A^(-1)C
(8)

and

A==ABB^(-1)==CB^(-1).
(9)

Therefore,

C==AB==(CB^(-1))(A^(-1)C)==CB^(-1)A^(-1)C,
(10)

so

CB^(-1)A^(-1)==I,
(11)

where I is the identity matrix, and

B^(-1)A^(-1)==C^(-1)==(AB)^(-1).
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