if z and r are both inverse of matrix a, prove that z=r

if z and r are both inverse of the matrix a, prove that z=r

since z and r are both inverses of a, we have az = za = i and ar = ra = i, where i is the nxn identity matrix (assuming that  a is nxn).
 
so, az = ar
 
multiply both sides on the left by r to get raz = rar
 
                                                           (ra)z = (ra)r
 
                                                               iz = ir
 
                                                                z = r
 

it follows from the methods used to find the inverse of a matrix that the result must be unique.
(each time that procedure is followed, there is one and only one result.)