A look at the vector a, should make this problem much easier to solve.
$\displaystyle [1, 0 \dots 0]^T \[ \begin{bmatrix} x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn} \end{bmatrix} \]$$\displaystyle \[ \begin{bmatrix}b_1 \\ \vdots \\ b_n\end{bmatrix} \]$
$\displaystyle = [1, 0 \dots 0]^T \[ \begin{bmatrix}b_1x_{11} + \dots b_nx_{1n} \\ \vdots \\ b_1x_{n1} + \dots + b_nx_{nn} \end{bmatrix}\]$
$\displaystyle = \[ \begin{bmatrix}b_1x_{11} + \dots + b_nx_{1n} \\ 0 \\ \vdots \\ 0 \end{bmatrix}\]$
So your objective function becomes $\displaystyle b_1x_{11} + \dots + b_nx_{1n}$
and your constraints are only the ones that apply to the variables in the objective function.
$\displaystyle x_{11} + \dots + x_{1n} = 1$
$\displaystyle 0 \leq x_{1i} \leq 1$ for $\displaystyle 1 \leq i \leq n$
I am not sure how you're supposed to solve the problem, but I'm pretty sure this set up is correct.