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 $i \ne j$.

Compute
$$\det\pmatrix{a&0\\0&1}$$
What do you notice?
The partition property is inherent because $\det$ is a function and $x \sim_f y :\Leftrightarrow f(x) = f(y)$ is an equivalence relation for any function $f$.
