Notation
If $A$ is a matrix with dimensions $(n,k)$ it is represented by $\hat A_{n,k}$
Background
Consider the following matrix equations:
$$ a \hat \kappa_{1 \times 2} + \hat b_{1 \times 2} \hat \kappa'_{2 \times 2} + e \hat \kappa''_{1 \times 2} = \lambda \hat \kappa_{1 \times 2}$$
$$ \hat b'_{2 \times 1} \hat \kappa_{1 \times 2} + \hat c_{2 \times 2} \hat \kappa'_{2 \times 2} + \hat d_{2 \times 1} \hat \kappa''_{1 \times 2} = \lambda \kappa'_{2 \times 2}$$
$$ e' \hat \kappa_{1 \times 2} + \hat d'_{1 \times 2} \kappa'_{2 \times 2} + f \hat \kappa''_{1 \times 2} = \lambda \kappa''_{1 \times 2} $$
The above can be neatly written as: $$ \begin{equation} \lambda \begin{bmatrix} \hat \kappa_{1 \times 2} \\ \hat \kappa'_{2 \times 2} \\ \hat \kappa''_{1 \times 2} \\ \end{bmatrix} = \begin{bmatrix} a & \hat b_{1 \times 2} & e \\ \hat b'_{2 \times 1} & \hat c_{2 \times 2} & \hat d_{2 \times 1} \\ e' & \hat d'_{1 \times 2} & f \\ \end{bmatrix} \begin{bmatrix} \hat \kappa_{1 \times 2} \\ \hat \kappa'_{2 \times 2} \\ \hat \kappa''_{1 \times 2} \\ \end{bmatrix} \end{equation}$$
$$ \begin{equation} \implies \begin{bmatrix} \hat 0_{1 \times 2} \\ \hat 0_{2 \times 2} \\ \hat 0_{1 \times 2} \\ \end{bmatrix} = \begin{bmatrix} a - \lambda & \hat b_{1 \times 2} & e \\ \hat b'_{2 \times 1} & \hat c_{2 \times 2} - \lambda . \hat I_{2 \times 2} & \hat d_{2 \times 1} \\ e' & \hat d'_{1 \times 2} & f - \lambda \\ \end{bmatrix} \begin{bmatrix} \hat \kappa_{1 \times 2} \\ \hat \kappa'_{2 \times 2} \\ \hat \kappa''_{1 \times 2} \\ \end{bmatrix} \end{equation}$$
where $\hat I$ is the identity matrix
Question
Is there any way one can define the determinant of such a matrix and find $\lambda$?