Consider the matrix:
$L=$\begin{bmatrix} x-5 && k && k && k&& k&& k\\k && x-5 && k && k&& k&& k\\k && k && x-5 && k&& k&& k\\k && k && k && x-4&& 0&& k\\k && k && k && 0&& x-3&& 0\\k && k && k && k&& 0&& x-4 \end{bmatrix}
The problem is I want to find the determinant of $L$ in terms of determinant of $L_{1,2,3}$ where $L_{1,2,3}$ is the matrix obtained by deleting the rows and columns corresponding to ${1,2,3}$.
Here $k\in \Bbb R$.
If $k=1$ then the solution is easy $\det L$ can be obtained by first multiplying the first row by $x-1$ and then applying the elementary row operations etc.
But I am having problem for $k\neq 1$
if $k\neq 1$ then $R_1=R_1(x-1)-R_2-R_3-\dots -R_n$
gives $R_1=$ $(x^2-6x+5(1-k), (k-1)(x-5),\dots ,(k-1)(x-5))$
and hence can't be expanded in terms of $L_{1}$ as in case of $k=1$.
Please give some hints on how to proceed in this case.