If A is 2 by 3 and B is 3 by 2 and AB I show from its rank

If A is 2 by 3 and B is 3 by 2 and AB = I, show from its rank that BA doesnt equal I. Give an example of A and B with AB = I. For m < n, a right inverse is not a left inverse.

Solution

Take Matrix
A= [ 1 0 0
1 1 1 ]

And matrix B= [ 1 0
-1 0
   0 1]   

A is a 2x3 matrix and B is a 3x2 matrix
Multiplying AB we get
AB= [ 1 0
0 1 ] Thus AB=I since it is an identity matrix of order 2

However when we multiply BA we get:

BA= [ 1 0 0
-1 0 0
   1 1 1 ] which is a 3x3 matrix but it isn\'t the identity matrix of order 3;

Thus, AB=I but BA is not equal to I;
Thus a right inverse is not a left inverse in matrices;