Trying to find the right answer to this..

Determinants

Question

If A, B, C are square matrices of the same order such that AB = BA = I and AC = CA = I then B = C.

Answer

We have
AB = BA = I    ...(1)
and AC = CA = I    ...(2)
Now B = BI
= B(AC)    [∵ of (2)]
= (BA) C    [∵ of associative properties of multiplication]
= IC    [∵ of (1)]
= C
∴ B = C
Hence the result.
Note : AB = BA = I ⇒ B is inverse of A
Again AC = CA = 1 ⇒ C is inverse of A
Also B = C
∴ inverse of a matrix A, if it exists, is unique.