Is it possible to get an example of two matrices Is it possible to get an example of two matrices $A,B\in M_4(\mathbb{R})$ both having $rank<2$ but $\det(A-\lambda B)\ne 0$ i.e it is not identically a zero polynomial. where $\lambda$ is indeterminate, I mean a variable. I want to say $(A-\lambda B)$ is of full rank matrix, assuming entries of $A-\lambda B$ is a polynomial matrix with linear polynomial.

Note that $rank\ A, B < 2$ means their ranks are 1 (If any of them is 0 your question has a trivial answer). That means that all columns of $A$ are $v, \alpha_v v, \beta_v v, \delta_v v$ and the columns of $B$ are $u, \alpha_u u, \beta_u u, \delta_u u$. Now note that the columns of $(A - \lambda B)$ will be sums of those multiples of $u$ with multiples of $v$. Regardless of how you distribute the $v, \alpha_v v, \beta_v v, \delta_v v$ with the $u, \alpha_u u, \beta_u u, \delta_u u$, all columns of $(A - \lambda B)$ will be of the form $xu + yv$ for some $x$ and $y$ scalars. Therefore, all your 4 columns will be a linear combination of the two vectors $u$ and $v$ and therefore the rank of $(A - \lambda B)$ can never be greater than 2.
