Question 376142
I'll change something in the problem a little bit:  let {{{alpha = a21/a11}}}.  then {{{(matrix(2,2, 1,0, alpha, 1))*(matrix(2,2,a11, a12,0,b)) = (matrix(2,2, 1,0, a21/a11, 1))*(matrix(2,2,a11, a12,0,b))  = (matrix(2,2,a11, a12, a21, (a12*a21)/a11 + b))}}}.  then {{{(a12)*(a21)/a11 + b = a22}}}, or {{{b = (a11*a22 - a12*a21)/a11 = det(A)/a11}}}.  Hence factorization of any 2x2 matrix is always possible in this format.