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:
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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)
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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.
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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.