When I was in school 55 years ago they taught us to do it this way. You can see that the determinant must be a polynomial of degree 6 in the variables a,b,c,d. If a=b then the first and second columns of the matrix are equal and so the determinant is 0. Therefore (by the factor theorem) is a factor of the determinant. For the same reason, , , , and are also factors. Thus the determinant is a multiple of . But since that is already a polynomial of degree 6, there cannot be any further factors apart from a numerical constant.

Thus . To find the numerical constant k, compare the coefficients of on both sides of that equation, and you find that . So .