Question

Calculate the eigenvalues and eigenvectors of a 5-by-5 magic square matrix

Answer 1

Consider an n × n matrix A and a scalar λ. By

definition λ is an eigenvalue of A if there is a

nonzero vector ~v in Rn such that

A~v = λ~v

λ~v − A~v = ~0

(λIn − A)~v = ~0

An an eigenvector, ~v needs to be a nonzero

vector. By definition of the kernel, that

ker(λIn − A) 6= {~0}.

(That is, there are other vectors in the kernel

besides the zero vector.)

Therefore, the matrix λIn−A is not invertible,

and det(λIn − A)=0.