Calculating the eigenvalues

Question

I'm trying to understand the dynamics of the eigenvectors and the eigenvalues. My question is about formula for finding the eigenvalues. At 4:15(the athor starts the calculation at 1:30) of the given video why should the determinant of the matrix be zero?

Thanks in advance.

Answer 1

We have an eigenvalue equation,

\begin{align*} Ax=\lambda x \end{align*}

where $A$ is a matrix, $x$ is an eigenvector, and $\lambda$ is the eigenvalue. This equation is the same as

\begin{align*} Ax-\lambda x=0\implies (A-\lambda I)x=0 \end{align*}

The goal is to find $x$ and $\lambda$. The only time when we get a nontrivial solution is when the determinant of $(A-\lambda I)$ is zero.