Trying to find the inverse of $B$ knowing the optimal solution

Question

Can one know how is $B^{-1}$ and $\left(\matrix{ b_1 \\b_2}\right)$ defined knowing that $c_BB^{-1}b=150$ and $B^{-1}b=B^{-1}$ $\left(\matrix{b_1\\ b_2 }\right)=\left(\matrix{30 \\ 10}\right)$ ?

We also know $c_B=[5,0]$

If it is possible, how to do that?

Answer 1

You can't uniquely determine $B^{-1}$ and $b$ in this setting.

For example, you can let $B^{-1}=I$ and $b=\begin{bmatrix} 30 \\ 10 \end{bmatrix}$.

Also, you can let $B^{-1}=2I$ and $b=\begin{bmatrix} 15 \\ 5 \end{bmatrix}$.