Romanian Matrix Problem

In this problem, I solve a neat matrix problem from a Romanian math competition, namely find all 2x2 matrices A such that A^3 - 3A^2 equals to the matrix whose entry is all -2. This is a beautiful application of linear algebra, more specifically traces and characteristic polynomials, enjoy! Note: This problem was originally featured on blackpenredpen's channel last year, here is the link: https://www.youtube.com/watch?v=rebbuTrZ--E Linear Algebra: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCM0HdpHbhWcKdM-oVXPPRI Subscribe to my channel: https://youtube.com/drpeyam Check out my TikTok channel: https://www.tiktok.com/@drpeyam Follow me on Instagram: https://www.instagram.com/peyamstagram/ Follow me on Twitter: https://twitter.com/drpeyam Teespring merch: https://teespring.com/stores/dr-peyam