The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. For a 2x2 matrix A with characteristic equation $lambda^2$ - (trace)lambda + det = 0, we get:

$A^2$ - (tr A)A + (det A)I = 0

Applications:

1. Finding A^(-1): From $A^2$ - (tr A)A + (det A)I = 0, multiply by A^(-1): A - (tr A)I + (det A)A^(-1) = 0, giving A^(-1) = [(tr A)I - A] / det(A)

2. Finding $A^n$: Reduce higher powers using the characteristic equation. E.g., $A^2$ = (tr A)A - (det A)I. Then $A^3$ = A * $A^2$ = A[(tr A)A - (det A)I] and substitute $A^2$ again.

3. Finding polynomial expressions: For f(A) where f is any polynomial, divide f(lambda) by the characteristic polynomial. The remainder (degree < n) gives f(A).

This is the fastest method for "Find $A^{100}$" type problems in JEE.