Find the minimal eigenvalue for the matrix A²⁰²¹

Below is a beautiful proposition on matrix eigenvalues.

It says find the minimal eigenvalue of the matrix $A^{2021}$.

We find the eigenvalue using the determinant for $|A-I\lambda|$ where $I\lambda$ is an identity $\lambda$ matrix. 

Next

=$(2-\lambda)[(-1-\lambda)(1-\lambda)]+0-0$

This yields a characteristic polynomial 

$=(2-\lambda)(-1+\lambda^2)$

Hence, our eigenvalues are 

$\lambda_1=\lambda_2=1,\lambda_3=-1$

$\min\{\lambda_i\}=-1$

And $A^{2021}=(-1)^{2021}=-1$