<s>[INST] Computing $K$-theory elements in a $C^*$ algebra $A$ Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. 
Now consider the matrices, the projections, 
$$
\left[ \begin{pmatrix}
1-p & 0 \\ 
0 & q
\end{pmatrix} \right] - \left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right] 
$$
does this define the same $K_0(A)$ element as $[p]-[q]$? 
 [/INST] I will assume you mean 
$$\left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix}
1-p & 0 \\ 
0 & q
\end{pmatrix} \right] ,
$$
since your first matrix is not a projection. Recall that 
$$
\left[\begin{pmatrix} p & 0 \\ 0 & 0 \end{pmatrix} \right]=\left[\begin{pmatrix} 0 & 0 \\ 0 & p \end{pmatrix} \right]=[p].
$$
Recall also that if $r,s$ are projections with $rs=0$, then $[r]+[s]=[r+s]$. Then 
$$
\left[\begin{pmatrix} 1-p & 0 \\ 0 & q \end{pmatrix} \right]
=\left[\begin{pmatrix} 1-p & 0 \\ 0 & 0 \end{pmatrix} \right]+\left[\begin{pmatrix} 0 & 0 \\ 0 & q \end{pmatrix} \right]=[1-p]+[q].
$$
So
\begin{align}
\left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix}
1-p & 0 \\ 
0 & q
\end{pmatrix} \right] 
&=\left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix}
1-p & 0 \\ 
0 & 0
\end{pmatrix} \right] -\left[ \begin{pmatrix}
0 & 0 \\ 
0 & q
\end{pmatrix} \right] \\ \ \\
&=\left[\begin{pmatrix} p+(1-p) & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix}
1-p & 0 \\ 
0 & 0
\end{pmatrix} \right] -\left[ \begin{pmatrix}
0 & 0 \\ 
0 & q
\end{pmatrix} \right] \\ \ \\
&=[p]+[1-p]-[1-p]-[q]\\ \ \\
&=[p]-[q].
\end{align}
</s>