Rank of a simple matrix series

Question

Problem Specifications and Given conditions

1. I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of $K_i, i>0 $ are less than 3.
2. I have another matrix with $M = K_0+\sum_{n=1}^{\infty}K_i s^n$. We know already rank of $K_0$ as 3. s is not a matrix,it is a variable
3. Dimension of $K_i$ and M are $ 3 \times 3$

Question

1. What is the rank of M? Can we say that it cant be less than 3?

2. What is the rank of $\sum_{n=1}^{\infty}K_i x^n$

Please provide proof thanks

Answer 1

In general no. You can say that the set of $s$ where the rank is maximal is open.

Let's say $M=M(s)$ and $L=M(1)$.

Consider $K_0=2I$, $K_1=\begin{pmatrix}-1&0&0\\0&0&0\\0&0&0\end{pmatrix}$ and $K_i=0$ for $i>1$.

Then the rank of $L=M(1)$ is $3$ but the rank of $M(2)$ is $2$.

$M(s)$ depends continuously on $s$. Therefore the set where $\det(M(s))\neq 0$ is open. So you only know that if $L$ has rank $3$, then this is true for $s$ sufficiently close to $1$.