I have following simplex problem $$\begin{array}{ll}\text{minimize} & z= -2x_1 - 3x_2 -6x_3\\\\ \text{subject to} &2x_1+x_2+x_3\le5\\ & 3x_2+2x_3\le6\\ & x_1, x_2, x_3 \geq 0\end{array}$$
I solved using simplex to get the following simplex table
$$\begin{matrix} \; & x_1 & x_2&x_3 &x_4 &x_5 &-z&b \\ x_1 & 1 & -1/4&0&1/2&-1/4&0 & 1 \\ x_3 & 0 &3/2 &1& 0 &1/2&0&3 \\ \hline -z&0&11/2&0 & 1& 5/2 &1 & 20 \end{matrix}$$
So I have optima $(1,0,3)$, I am further asked in the problem if this optima is unique? How does one show that optima given by simplex method is unique.
My intuition is that at each step there was only one possible variable to include in the new feasible solution, hence the obtained optima should be unique. Is this correct? If yes then how does one put it mathematically? If no then what is the method to think about this?