Q:

AB = I and BC = I then what is the relation between A and C?

B = matrix given below. I is identity matrix.
 [1 2 3 4]
 [3 2 4 3]
 [1 3 2 4]
 [5 4 3 7]

So What will be the relation between the matrices A and C if AB = I and BC = I?
I think that A = C because both AB and BC have B in common and both of their product is an identity matrix but I'm not sure. 

A:

Suppose $AB=I$ and $BC=I$; then
$$
A=AI=A(BC)=(AB)C=IC=C
$$

