Question

Let λ be an eigenvalue of the matrix A. show that 2lambda is an eigenvalue of the matrix 2A.

Answer 1

If λ is the eigenvalue of the matrix A, then we can write
Av = λv for a non-zero matrix v.

now we need to find the eigenvalue of 2A.
Multiply 2 to both sides
2 × Av = 2 × λv
⇒2Av = 2λv
⇒(2A)v = (2λ)v

So for a non-zero matrix v, we can write (2A)v = (2λ)v.
So the eigenvalue of 2A is 2λ.