To understand Caley-Hamilton theorem you have to know about characteristic polynomial.
Characteristic Polynomial
If A is a square matrix of of size n * n, and I is identity matrix of the same size as A. Then determinant |A-λI| will result in an equation of the form.
Which is called characteristic equation of matrix A
The root of characteristic equation are called eigen values of matrix A.
Caley-Hamilton theorem
The theorem states that characteristic equation of a square matrix is satisfies by itself.
i.e.
I will explain it using a 2 × 2 matrix
Conclusion-
Theorem is very important result from linear algebra. Which is a lot of applications in matrix and polynomial related problem. In this post, I have explained the theorem in a very concise form. Hope it will help you for further study of linear algebra and, further, you will also apply somewhere.