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