Given two 3*4 matrix M and M', M = [A,b], M' = [A', b']. A and A' are 3*3 invertible matrix, b and b' are 3*1 matrix.

Also assume e^t(-A^TA^-1b + b') !=0, where e^t = [0,0,1]^t.

M_hat = [I_3*3, 0_3*1], M_hat' = [m1_1*4, m2_1*4, [0,0,0,1]] where m1 and m2 are any 1*4 matrix.

Show that there is a 4*4 matrix H such that MH = M_hat and M'H = M_hat'.

Hint is to decompose H into H0 and H1 such that MH0 = M_hat. After find H0, then find H1, so that M'H0H1 = M_hat' and also keep MH0H1 = M'.