Question

The scalar 2. is an Eigen value of the matrix A if
(a) (A - AI) is non-singular (b) (A - Z) is singular
(D) Ais Singular
(C) A is Non-Singular​

Answer 1

We know that 0∈λ(A) iff there exists some nonzero solution to the eigenvector equation Ax=λx=0⋅x=0. Thus 0 is an eigenvalue iff ∃b∈Ker(A) with b≠0. But since Ker(A)≠{0} we conclude that A must be singular.