Question

The geometric multiplicity of λ is its multiplicity as a root of the characteristic polynomial of A, where λ be the eigen value of A.

(a) True

(b) False

This question was addressed to me in a national level competition.

My doubt stems from Diagonalization Powers of a Matrix topic in chapter Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

Correct option is (b) False

For explanation I would say: The diagonalization theorem in terms of multiplicities of eigen values is defined as follows,

The algebraic multiplicity of λ is its multiplicity as a root of the characteristic polynomial of A.

The geometric multiplicity of λ is the dimension of the λ-eigenspace.