Invertible matrix sum

Question

I have a matrix $\mathbf{T}=\left[ \mathbf{t}_0 + \mathbf{A}_0\mathbf{x},\mathbf{t}_1 + \mathbf{A}_1\mathbf{x},\cdots,\mathbf{t}_n + \mathbf{A}_n\mathbf{x}\right]\in\mathbb{R}^{n \times n }$ where $\mathbf{t}_i\in\mathbb{R}^{n \times 1}$ and $\mathbf{A}_i\in\mathbb{R}^{n \times p}$ are given column vectors, resp. matrices. The vector $\mathbf{x}\in\mathbb{R}^{p \times 1}$ has to be determined such that that the matrix $\mathbf{T}$ is invertible. I know that it will not be possible in general. In this case I know by previous results that a solution must exist. Is there a simple way (e.g. without optimization) to specify $\mathbf{x}\in\mathbb{R}^{p \times 1}$? E.g. series expansion? Thank you very much in advance.