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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.

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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.

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  • $\begingroup$ Is it because the matrix must have no inverse? $\endgroup$
    – Snate
    Jul 6, 2016 at 21:50
  • $\begingroup$ @Snate If $(A - \lambda I)$ is invertible then the only solution to $(A- \lambda I)x = 0$ is $x = 0$, which is precisely what we want to exclude. $\endgroup$ Jul 6, 2016 at 22:38

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