Question

Suppose that λ is an eigenvalue of A. Show that 2λ is then an eigenvalue of 2A.

Answer 1

Proof:

If λ is an eigenvalue of A then for some x≠0,

`Ax=\lambda x\\\\Adding\,\,Ax\,\,on\,\,both\,\,sides\\\\Ax+Ax=\lambda x+Ax\\\\2Ax=\lambda x+\lambda x\\\\2Ax=2\lambda x`

Which shows that if λ is an eigenvalue of A, 2λ is then an eigenvalue of 2A.