Identity matrix in rewritten eigenvalue definition

Question

I see that an eigenvalue eigenvector pair is described as satisfying the equation $Ax = \lambda x, x \neq 0$

I also see that this equation can be rewritten as $(\lambda I - A)x = 0, x \neq 0$

I'm not sure what intermediate steps were taken to get from the first equation to the second and why there is the sudden appearance of the identity matrix multiplied by the scalar eigenvalue $\lambda$.

Could someone enlighten me if it isn't too much trouble?

Answer 1

Let $A$ be a $n \times n$ matrice. Note that $Ix = x$ because $x = (x_{1},...,x_{n})$ and $I$ is a $n \times n$ matrice. So, $$Ax = \lambda x \Longleftrightarrow Ax = \lambda Ix \Longleftrightarrow \lambda Ix - Ax = 0 \Longleftrightarrow (\lambda I - A)x = 0.$$