Isometry and determinant of matrix

Let e_1 = (1,0) and e_2 = (0,1). If T is an isometry of the plane fixing O, let T(e_1) = (a,b) and

T(e_2) = (c,d) and let A be a 2x2 matrix such that the first row is (a c) and second row is (b d). Prove that det(A) = 1 or -1.

Using the definition of isometry i obtained the following equations:

a^2+b^2=1 ----------(1)

c^2+d^2=1 ----------(2)

ac+bd=0 -------------(3)

However, i was unable to continue on to find det(A). Can this problem be solved just by using definition of isometry alone? Or do i need some results from linear algebra. (This problem was part of a group theory course)

Re: Isometry and determinant of matrix

Re: Isometry and determinant of matrix