# 2 by 2 matrices with a fixed determinant

Fix a complex number $$z$$ with modulus at most one. What are all examples of non-triangular $$2\times 2$$ matrices with $$z$$ as its determinant? The question has satisfactory answer in the triangular matrix situation because of two reasons: (1) every matrix can be (say) upper triangularized and (2) for the triangular matrices, the question boils down to finding all possible factorizations $$z=\lambda_1\cdot\lambda_2$$ and then the matrices of the following form are all: $$\begin{bmatrix} \lambda_1 & *\\ 0 & \lambda_2 \end{bmatrix}$$ This is why I am interested in the non-triangular case. Therefore I am interested in knowing all possible matrices of the form $$\begin{bmatrix} x_1 & x_2\\ x_3 & x_4 \end{bmatrix}$$ such that none of $$x_2,x_3$$ is zero and that $$x_1x_4-x_2x_3=z.$$

If $$x_3 \neq 0$$, then we can solve for $$x_2$$ as follows: $$x_1x_4 - x_2x_3 = z \implies x_2 = \frac{x_1x_4 - z}{x_3}$$ If you like, we could parameterize the set of matrices for which $$x_2,x_3$$ are non-zero as follows: $$M = \pmatrix{a& \frac{ac - z}{b}\\b&c} \quad a,b,c \in \Bbb C, \quad b \neq 0, \quad ac \neq z$$