The matrix A = [1 2; 3 6] is inconsistent for certain choices of vector b in R2. However, the associated equation ATAx = ATb is consistent for any choice of b. A solution x* to ATAx = ATb can be found. For any random vector x in R2, the inequality ||Ax* - b|| ≤ ||Ax - b|| holds. This means that x* minimizes the length of the residual vector r = Ax - b.