2 by 2 matrices with a fixed determinant

Question

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 $x_1x_4-x_2x_3=z.$

Answer 1

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: $$ M = \pmatrix{a& \frac{ac - z}{b}\\b&c} \quad a,b,c \in \Bbb C, \quad b \neq 0, \quad ac \neq z $$