The Relations Between Kernel, Rank and Geometric Multiplicity

Question

For the sake of simplicity I'll give an example: Let A be a square real square matrix of order 4 and:

kerA = 3
rankA = 1
λ=1 is an eigenvalue

What can we know about its eigenvalues?

So far I've figured out 0 must be an eigenvalue since A is singular, but I'm struggling with its geometric and algebraic multiplicity.

Answer 1

If the kerA = 3, then there are 3 linearly independent vectors that are eigenvectors with eigenvalue A (because Av=0v for all v in the kernel). So the geometric multiplicity of 0 is 3.