Looks like there are multiple determinants that represent an expression.
In[3]:= Reduce[
Det[{{a1, b1}, {c1, d1}}] == Det[{{a, b}, {c, d}}], {a1, b1, c1, d1}]
Out[3]= (a1 != 0 && d1 == (-b c + b1 c1 + a d)/a1) || (a1 == 0 &&
b1 != 0 && c1 == (b c - a d)/b1) || (d == 0 && c == 0 && a1 == 0 &&
b1 == 0) || (d != 0 && a == (b c)/d && a1 == 0 &&
b1 == 0) || (d == 0 && c != 0 && b == 0 && a1 == 0 && b1 == 0)
