## elements of $K_0$ group

Let $e$ be an idempotent in a unital algebra $A$ and $u=\begin{pmatrix}1-e & e\\ e & 1-e\end{pmatrix}$, then $u$ is invertible and $u\begin{pmatrix}1-e & 0\\ 0 & e\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}u.$ So, $\begin{pmatrix}e & 0\\ 0 & 1-e\end{pmatrix}$ is Murrary-von Neumann equivalent to $\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$ in $M_2(A)$.

Suppose $g=[e]-[f]$ is an element of a $K_0$ group, then \begin{align*}g=&[e]+[1-f]-[1-f]-[f]=[\begin{pmatrix}e & 0\\ 0 & 1-f\end{pmatrix}]-[\begin{pmatrix}1-f & 0\\ 0 & f\end{pmatrix}]\\=&[\begin{pmatrix}e & 0\\ 0 & 1-f\end{pmatrix}]-[\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}].\end{align*}