Linear Algebra Question for Repeated Eigenvalues

Question

Can somebody explain what the sentence in the red circle means?

Answer 1

Intuitively it means that, when combined with the other equations, they provide the same information. More rigorously, it means that there is some "linear combination" of the equations which gives $0=0$.

This is easiest to see when, as in your case, you have $n$ equations and $n-1$ of them are redundant; this means that each one is a scalar multiple of the other.

Answer 2

Since $\det(A-\lambda I) = 0$, $A-\lambda I$ is singular and therefore not full (row) rank. Therefore some equations can be written into a linear combination of others, and those equations are considered to be "redundant", as they give no additional information about the solutions.

Answer 3

It means that you don't need them to solve the system. The other equations gave you already all the information you needed, thus they are redundant.