Prove that the inverse of a square matrix, if it exists, is unique.

1.	Prove that the inverse of a square matrix, if it exists, is unique.
Answer» Let A be a square matrix. If possible let B and C are its two inverses. As B is the inverse of A AB = BA = I …(1) As C is the inverse of A AC = CA = I …(2) B = BI = B(AC) = (BA)C = IC = C B = C Hence the inverse of A is unique.

Answer»

Let A be a square matrix. If possible let B and C are its two inverses.

As B is the inverse of A

AB = BA = I …(1)

As C is the inverse of A

AC = CA = I …(2)

B = BI = B(AC) = (BA)C = IC = C

B = C

Hence the inverse of A is unique.

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Prove that the inverse of a square matrix, if it exists, is unique.

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