Equivalence class for a determinant

Question

Show that the relation on the set of all 2 x 2 matrices defined by A ~ B if detA=det B is an equivalence relation. Describe the equivalence class.

I have determined that it is an equivalence relation. I am troubled by the equivalence class part. Determinants exist in the set of real numbers, right? How would I indicate the matrices that satisfy this?

Answer 1

You say, matrix is of order $2\times 2 $.

Det can be any complex number.

Fix a complex number $c$.

Now,,take the matrix $$ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{pmatrix} $$

Now , for the equivalence class of $c$ , $a_{11} a_{22} - a_{12} a_{21}= c $

There infinitely many $a_{11} ,a_{22},a_{12} , a_{21} $ satisfying the above equation.

So,, there are uncountably many class and each class contains uncountably many matrices.

Answer 2

It's clear that the relation $A\sim B$ is an equivalence relation :

$\textbf{Reflexive:}$ $A\sim A$ is clear because $$det A=det A$$

$\textbf{Symmetric:}$ If $A\sim B$ then $B\sim A$ since $$A\sim B \iff det A= det B \iff det B= det A\iff B\sim A$$

$\textbf{Transitive:}$ Let $A$,$B$,$C$ be three matrices :

$A\sim B$ and $B \sim C$ $\implies$ $det A= det B$ and $det B = det C$

$\implies det A=det C \implies A\sim C$

So it's an equivalence relation.

For the equivalence class let $detA=z\in \mathbb C$, we have :

$$\begin{align} \bar{A} &=\{B\in M_2(\mathbb C) / A\sim B\}\\ &=\{B\in M_2(\mathbb C) / det A= det B\}\\ &=\{B\in M_2(\mathbb C) / a_1a_4-a_3a_2=b_1b_4-b_3b_2=z\}\\ \end{align} $$

Which gives $ b_1b_4-b_3b_2=z$, which has infinitely many solutions.

So we can only write

$$\bar{A}=\{B\in M_2(\mathbb C) / det B=z\}$$