Home | 18.022 | Chapter 15 | Section 15.3

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For every element v of the vector space consider the pair of vectors (v, Av). Suppose you have A-1A = In.
Since A transforms v to Av, then  A-1 must transform Av back to v. But then A-1 followed by A takes Av to v and back again to Av, so that you also have AA-1 = In, at least in its action on vectors that can be written in the form Av.
In finite dimensions every vector is in the range of a non-singular matrix transformation, so that either equation implies the other.