Proof: The identity matrix is invertible and the inverse of the identity is the identity

Question

How can i show that: $II^{-1} = I = I^{-1}I$ (the identity matrix is invertible) for all cases. And then proof that: $I^{-1} = I$ (The inverse of the identity is the identity). I don't know how start both proof, any suggestion?

Answer 1

Suppose $A$ is the inverse of the identity matrix. Then $AI =IA = I$. But $AI = IA = A$ as well so $A=I$.

Answer 2

$II=I$, so immediately $I=I^{-1}$. This completes both of your proofs.

Answer 3

AS INVESE OF ANY MATRIX IS GIVEN BY A FORMULA; A-1=[1/|A|]X[ADJ OF A] BUT HERE WE PROOF BY GENERAL METHOD. LET K IS INVERSE OF IDENTITY MATRIX I THEN WE KHOW THAT AS, KI=IK=I
ALSO,KI=IK=K SO,I=K OR [I=I-1]

SO INVERSE OF IDENTITY MATRIX IS IDENTITY MATRIX.

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