DISCLAIMER: This problem is explained properly on my YouTube channel, at the following link: PROBLEM: Analysis of 2x2 Matrices using Cayley-Hamilton Theorem. I suggest you watch this before looking below.
Before I start, let's recall the Cayley-Hamilton Theorem:
Theorem (Cayley-Hamilton Theorem) If is a square matrix and is its characteristc polynomial, then (allowing for matrices as the polynomial variable) , where is the zero square matrix.
Suppose now that is the following 2 x 2 matrix:
Note that the trace (denoted by ) and determinant (denoted by ) is and respectively. Then the characteristic polynomial of is:
Thus, by the Cayley-Hamilton Theorem:
What we can see is that can be expressed in terms of , and . Can we express in terms of , and ? Well, let's compute this:
So here's two open problems for you:
1) How to express in terms of , and ?
2) Can we do something similar for square matrix of arbitrary sizes? How about a 3 x 3 matrix?