Cayley-hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation, a powerful JEE Advanced tool for reducing matrix powers. For A with rows (2,1),(1,2), the trace is 2 + 2 = 4 and the determinant is 2·2 - 1·1 = 3. The characteristic equation of a 2x2 matrix is lambda^2 - (trace)lambda + (det) = 0, namely lambda^2 - 4 lambda + 3 = 0. By Cayley-Hamilton, A itself satisfies A^2 - 4A + 3I = O, so A^2 = 4A - 3I. The scalar multiplying the identity is therefore 3, which is exactly the determinant. Option 4 confuses the scalar with the trace coefficient. Option 5 has no basis in the characteristic equation. Option 1 ignores the determinant value. Hence the scalar is 3. Plausibility check: computing A^2 directly gives rows (5,4),(4,5), and 4A - 3I gives (8-3, 4),(4, 8-3) = (5,4),(4,5), matching exactly and confirming A^2 = 4A - 3I. The Cayley-Hamilton theorem lets every power of a matrix be rewritten as a linear combination of lower powers, which is the standard device for computing high powers and inverses efficiently. For a two by two matrix the characteristic relation immediately expresses the square in terms of the matrix and the identity, and iterating it expresses all higher powers without ever multiplying matrices directly.