Eigenvalue of altered matrix: $pI_n + qA$

Question

As a part of an exercise I have to prove the following:

Let $p,q \in \mathbb{R}$. Let $A$ be an $(n \times n)$ matrix. Let $I_n$ be the $(n \times n)$ identity matrix.

If $A$ has an eigenvalue $\lambda$, then $pI_n + qA$ has a corresponding eigenvalue $\lambda' = p + q \lambda$.

How can I prove this?

Thanks and regards.

Answer 1

$Ax=\lambda x\Rightarrow (pId+qA)x=px+qAx=px+q\lambda x=(p+q\lambda)x$