Suppose a determinant $A$ is given by:
$ \begin{vmatrix} f(x) & g(x) & a_1 \\ h(x) & a_2 & j(x) \\ a_3 & k(x) & a_4 \end{vmatrix} $
where $f(x), g(x), h(x), j(x), k(x)$ are some functions of $x$ and $a_1$,$a_2$, $a_3$, $a_4$ are some constants such that when $x=a$, exactly two columns and two rows become identical.
If we substitute $x = a$, we get two columns (say column 1 and 2) as well as two rows (say row 1 and 2) identical. Then what will be the power of $x-a$ in the factorisation of the given determinant $A$?
Example: I am not able to find a reference but I have come across that when $r$ rows become identical when $x=a$ is substituted, then the power of $x-a$ in the factorisation of the determinant is equal to $r-1$. I want to prove if this applies when $r$ rows and $c$ columns become identical when $x-a$.
Let us assume wlog $a=0,$ and examin more generally some $3\times3$ determinant where all entries are polynomials, $$B:=\begin{vmatrix} f(x) & g(x) & u(x) \\ h(x) & v(x) & j(x) \\ w(x) & k(x) & i(x) \end{vmatrix},$$ such that when we substitute $x=0,$ the 1st and 2nd columns become equal, and the 1st and 2nd rows become equal. This is equivalent to: $$g(x)-f(x)=xp(x),\quad v(x)-h(x)=xq(x),\quad k(x)-w(x)=xr(x),$$ $$h(x)-f(x)=xs(x)\quad\text{and}\quad j(x)-u(x)=xt(x)$$ for some polynomials $p(x),q(x),r(x),s(x),t(x).$
(We already know from this that $v(x)-g(x)$ is also a multiple of $x$: $v(x)-g(x)=x\left(q(x)+s(x)-p(x)\right)$.
Now, by the properties of the determinant, substracting the 1st column from the second one and then the first row from the second one, we get: $$B=\begin{vmatrix} f(x) & xp(x) & u(x) \\ h(x) & xq(x) & j(x) \\ w(x) & xr(x) & i(x) \end{vmatrix}=\begin{vmatrix} f(x) & xp(x) & u(x) \\ xs(x) & xe(x) &xt(x) \\ w(x) & xr(x) & i(x) \end{vmatrix}$$ where $e(x):=q(x)-p(x).$
Expanding the determinant, we get: $$B=x\begin{vmatrix} f(x) & p(x) & u(x) \\ xs(x) & e(x) &xt(x) \\ w(x) & r(x) & i(x) \end{vmatrix}\equiv xe(x)\Bigl(f(x)i(x)-w(x)u(x)\Bigr)\bmod{x^2}.$$
Therefore, $0$ is a generally a root of $B$ of multiplicity only $1$ (like when the hypothesis was only about two rows or two columns but not both). However it has a multiplicity $\ge2$ if $f(0)i(0)=w(0)u(0)$ or $e(0)=0.$
As a particular case, $a$ is generally a root of $A$ of multiplicity only $1,$ but has a multiplicity $\ge2$ if $a_2a_4=a_3a_1$ or if it is a multiple root of $a_2-h(x)-g(x)+f(x).$