give example of a matricesuch that A AT = I but AT A does not equal to I - Maths - Matrices

Question

give example of a matricesuch that A AT = I but
AT A does not equal to I

Anuradha Sharma · Accepted Answer

A = abcdAT =acbd Now AAT= abcdacbd  =a2+b2ac+bdac+bdc2+d2 =1001 comparing we get a2+b2=1Now from here either a=0 and  b= 1 or  a= 1 and b = 0 
1ac+bd=0
2c2+d2=1from here either c=0 and  d= 1 or  c= 1 and d = 0 
3Putting this in equation 2, by considering first part of equation1 only, 0+d=0d = 0 So , second passibility of equation 3 is correct
Hence we get, A =0110 and with the second possibility, A =1001 Now we have shown that A= 0110 or 1001 would show that A AT= IBut there is not possible case that ATA≠I when A AT= I

give example of a matricesuch that A .AT = I but AT.A does not equal to I

give example of a matricesuch that A .A^{T =}I but A^T.A does not equal to I