Show that ~ creates a partition of $M_2(\mathbb{R})$ Let $M_2 (\mathbb{R})$ be the set of 2x2 matrices over $\mathbb{R}$: 
$$
M_2 (\mathbb{R}) = \biggl\{ 
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\; \biggm| \; \text{with } a,b,c,d \in \mathbb{R} \biggr\}. 
$$
For $M_1, M_2 \in M_2(\mathbb{R})$, we say that $M_1 \sim M_2$ if $\det(M_1) = \det(M_2)$, where 
$$
\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad-bc.
$$


*

*Show that $\sim$ creates a partition of $M_2(\mathbb{R})$. 

*What are the representative elements of each partition?

*Are there countably or uncountably many distinct equivalence classes?


I am stuck and I could really use some help. I know that in order to have a partition, the union of all the partitions $X_i$ should be $X$ and $X_i \cap X_j = \varnothing$ for $i \ne j$.

Hints:


*

*It suffices to show that $\sim$ is an equivalence relation on $M_2(\mathbb R)$. So simply show that it's reflexive, symmetric, and transitive. This follows immediately, since $\sim$ is defined in terms of $=$.

*Try some examples. Pick a matrix at random, and find its equivalence class. Then try going backwards: pick a determinant at random, then try to find a matrix that has that as a determinant.

*From the above bullet, try to show that each $x \in \mathbb R$ corresponds to a distinct equivalence class $[M]_\sim$.
