To Prove: If

$$A_{nxn}$$

and

$$B_{nxn}$$

such that AB=I, then BA=I

$$AB=I \implies Rank(A), Rank(B)=n$$

Reason: Rank(AB)

$$\leq$$

min(Rank A, Rank B)

so B is a full rank matrix. Now consider B=BI

$$\implies$$

B-B(AB)=0

$$\implies$$

B-(BA)B=0

$$\implies$$

(I-BA)B=0

Since B is full rank so I-BA=0. Q.E.D

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