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: $lambda^3$ - p*$lambda^2$ + q*lambda - 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 $A^3$ = 5$A^2$ - 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)"