The Cayley-Hamilton theorem states: every square matrix satisfies its own characteristic equation.

For 2x2 matrix A: Characteristic equation: lambda^2 - (tr A)lambda + det(A) = 0 Cayley-Hamilton: A^2 - (tr A)A + (det A)I = O

For 3x3 matrix A: Characteristic equation: lambda^3 - plambda^2 + qlambda - r = 0 where p = tr(A), q = sum of 2x2 principal minors, r = det(A) Cayley-Hamilton: A^3 - pA^2 + qA - rI = O

Application 1: Finding A^(-1) From A^2 - (tr A)A + (det A)I = O: A^(-1) = [(tr A)I - A] / det(A)

Application 2: Finding A^n Express higher powers in terms of lower powers using the characteristic equation. Example: If A^2 = 5A - 6I, then A^3 = 5A^2 - 6A = 5(5A-6I) - 6A = 19A - 30I

Application 3: Matrix polynomials To find f(A) for any polynomial f, divide f(lambda) by the characteristic polynomial. The remainder (degree < n) gives f(A) directly.

JEE Tip: Cayley-Hamilton is the fastest method for:

• "Find A^(-1)" when trace and determinant are easy to compute
• "Find A^100" or similar high-power expressions
• "If A^2 - 5A + 6I = O, find A^3, A^4, or A^(-1)"