Unitary Matrix Representation for SU(2) Group: Derivation and Verification

[spaghetti3451]

The matrix representation $U$ for the group $SU(2)$ is given by

$U = \begin{bmatrix} \alpha & -\beta^{*} \\ \beta & \alpha^{*} \\ \end{bmatrix}$

where $\alpha$ and $\beta$ are complex numbers and $|\alpha|^{2}+|\beta|^{2}=1$.

This can be derived using the unitary of $U$ and the fact that $\text{det}\ U=1$.Is any complex $2\times 2$ matrix with unit determinant necessarily unitary?Consider the following argument:

$\text{det}\ (U) = 1$
$(\text{det}\ U)(\text{det}\ U) = 1$
$(\text{det}\ U^{\dagger})(\text{det}\ U) = 1$
$\text{det}\ (U^{\dagger}U) = 1$
$\text{det}\ (U^{\dagger}U) = \text{det}\ (U)$
$U^{\dagger}U = U$
$U^{\dagger}= 1$

Where's my mistake in this argument?

 

[TeethWhitener]

$\det(A) = \det(B)$ does not imply $A=B$.

 

[spaghetti3451]

Thanks!

I was wondering what is the most general form of the complex $2 \times 2$ matrix with unit determinant.

My first hunch was that it is the $2\times 2$ matrix representation of the $SU(2)$ group, but then, a complex $2 \times 2$ matrix with unit determinant is not necessarily $SU(2)$.

Can you help me with finding the most general form of the complex $2 \times 2$ matrix with unit determinant?

 

[TeethWhitener]

Can you help me with finding the most general form of the complex 2×22×22 \times 2 matrix with unit determinant?

I don't think there's anything special about it. As far as I know, it's just:
$$U= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$
where $a,b,c,d \in \mathbb{C}$ and $ad-bc = 1$.