# Solve Determinant Equation

Identify $$t\in\mathbb{R}$$ such that (basically it is a determinant of a block matrix of size $$n$$): $$\left| \begin{array}{cc} \mathbf{x}+t\mathbf{y} & \mathbf{B} \\ \mathbf{c} & \mathbf{D} \end{array} \right| = 0$$

where $$\mathbf{x}\in\mathbb{R}^{m\times1}$$, $$\mathbf{y}\in\mathbb{R}^{m\times1}$$, $$\mathbf{c}\in\mathbb{R}^{(n-m)\times1}$$ are column vectors, and $$\mathbf{B}\in\mathbb{R}^{m\times (n-1)}$$, $$\mathbf{D}\in\mathbb{R}^{(n-m)\times (n-1)}$$ are matrices, and $$m$$ is an integer with $$1 \le m \le n$$, and $$t\in\mathbb{R}$$ is the unknown variable we want to solve.

I am not sure if the solution exists in general cases, but in my cases, solution $$t$$ exists.

I encountered such a problem when I want to write a computer program that finds a proper value $$t$$ that makes some points coplanar, such a geometric problem can be finally reduced to the above mathematical problem after some calculation.

The problem is quite challenging, because I would like to find a closed-form solution such that $$t=...$$ (right hand side is composed of known variables) so that computer can compute the solution $$t$$, otherwise we have to use symbolic calculation which is unfriendly to a computer. Even an approximate approach (e.g. iterative numerical approach) is also acceptable if the exact solution is too hard to find.

Do you have any thoughts on this? I would be grateful if you could provide some hints on this issue.

Let $$X:=\begin{pmatrix} \mathbf{x}& \mathbf{B} \\ \mathbf{0}& \mathbf{D} \\ \end{pmatrix}$$ and let $$Y:=\begin{pmatrix} \mathbf{y}& \mathbf{B} \\ \mathbf{c}& \mathbf{D} \\ \end{pmatrix}$$. By multilinearity of the determinant it holds: $$0=\det\begin{pmatrix} \mathbf{x}+t \mathbf y& \mathbf{B} \\ \mathbf{c}& \mathbf{D} \\ \end{pmatrix}= \det X + t \det Y \Leftrightarrow t\det Y=-\det X$$
So if $$\det Y=\det X=0$$, any $$t\in \mathbb R$$ works; if $$\det Y=0\neq \det X$$, no $$t\in \mathbb R$$ works; if $$\det Y\neq 0$$, then $$t=-\frac{\det X}{ \det Y}$$.