Is the inverse of a matrix unique?? If yes how do we prove that ?
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Assume that a nonsingular matrix has two inverse matrices and then show that the two matrices are equal to each other.
The same proof to show inverses in a group are unique works. Let $\displaystyle A_1^{-1}$ and $\displaystyle A_2^{-1}$ be inverses of A. $\displaystyle A_1^{-1}=A_1^{-1}I_n=A_1^{-1}(AA_2^{-1})=(A_1^{-1}A)A_2^{-1}=I_nA_2^{-1}=A_2^{-1}$
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