Can the 'Determinant' be Reversed for 2-by-2 Matrices?

[scikidus]

Let there be a 2-by-2 matrix A with the elements:

[a b]
[c d]

Now, let there be a 2-by-2 matrix B with the elements:

[w x]
[y z]

Let A*A = B.

This means that w, x, y, and z can all be independantly represented solely in terms of a, b, c, and d.

My question: is there any way for a, b, c, and d to be represented solely in terms of w, x, y, and z?

 

[HallsofIvy]

Well, multiplying A*A out, you have a²+ bc= w, ab+ bd= x, ac+ cd= y, and bc+ d²= z. Now it is a matter of solving those 4 equations for a, b, c, and d.
Those are quadratic equations so there will be more than one solution - as you might expect from the fact that A*A= B is really "quadratic" itself.

Now, my question is, What does this have to do with the "determinant"?