Given the following linear program:
\begin{cases}
\max &5x_1 + x_2 + 6x_3 + 2x_4\\
&4x_1 + 4x_2 + 4 x_3 + x_4\le 24\\
&8x_1 + 6x_2 + 4x_3 + 3x_4\le 36\\
&\forall i, x_i\ge 0
\end{cases}
How to know if the basis $B=\{3,4\}$ is an optimal basis?
I only found that it is a feasible basis as far as first
$$B=
\begin{pmatrix}
4 & 1\\
4 & 3
\end{pmatrix}$$
is invertible becasuse $\det B =8 \neq 0$. And second because:
$$B^{-1}b=
\frac{1}{8}
\begin{pmatrix}
3/8 & 1/8\\
-1/2 & 1/2
\end{pmatrix}
\begin{pmatrix}
24\\36
\end{pmatrix}\ge 0
$$
Best Answer
You need to check the reduced costs. The basis is optimal if the reduced costs are negative (for a maximization problem).
The basis is $\{3,4\}$, so the non basic variables are $x_1$ and $x_2$.
$x_1$ has reduced cost $$ c_1-z_1=5-6(3/64)-2(-1/16) = 155/32 >0, $$ so if $x_1$ enters the basis, the objective function will increase by $155/32$ for every unit of $x_1$: the basis $\{3,4\}$ is not optimal.