Let $m>0$ be an integer. Given non-zero real numbers $b_1, \ldots, b_m$, can we always find real numbers $a_{i,j}, i=1,\ldots,m-1$, $j=1,\ldots, m-1$ such that
$$\mathbb \det \begin{pmatrix} a_{1,1} & \ldots & a_{1,m}\\ \vdots & & \vdots \\ a_{m-1,1} & \ldots & a_{m-1,m}\\ \sum\limits_{i=1}^{m-1} a_{i,1} b_i & \ldots & \sum\limits_{i=1}^{m-1} a_{i,m} b_i\\ \end{pmatrix} \neq 0$$
If so, how can we prove that ? And could we relax the restrictions on $b_i$ (all non-zero) ?
The matrix will never be invertible as the last row is a linear combination of the others. This also applies to the case $m=1$ in which case we simply have the zero matrix (because the sum is then empty).
And if the matrix is not invertible its determinant will be $0$