oxidricasbt7

2022-12-20

Eigenvalues of a $2×2$ matrix A such that ${A}^{2}=I$
I have no idea where to begin.
I know there are a few matrices that support this claim, will they all have the same eigenvalues?

Kendall Cortez

Expert

If v is an eigenvector of A with eigenvalue $\lambda$, then
$v=Iv={A}^{2}v=A\left(Av\right)=A\left(\lambda v\right)=\lambda \left(Av\right)={\lambda }^{2}v.$
Thus, if $\lambda$ is an eigenvalue of A and ${A}^{2}=I$ then ${\lambda }^{2}=1$. This gives only two possibilities for $\lambda$, $±1$.
Notice, that we never assumed that A is $2×2$. Indeed, if A is any square matrix and ${A}^{2}=I$ then the only possible eigenvalues of A are $±1$.

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